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(************************************************************************) (* * The Coq Proof Assistant / The Coq Development Team *) (* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) (* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) (* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import Rbase. Require Import Ranalysis_reg. Require Import Rfunctions. Require Import Rseries. Require Import Lra. Require Import RiemannInt. Require Import SeqProp. Require Import Max. Require Import Omega. Require Import Lra. Local Open Scope R_scope.
forall (f g : R -> R) (lb ub : R), lb < ub -> (forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f y) -> (forall x : R, f lb <= x -> x <= f ub -> comp f g x = id x) -> (forall x : R, f lb <= x -> x <= f ub -> lb <= g x <= ub) -> forall x y : R, f lb <= x -> x < y -> y <= f ub -> g x < g yforall (f g : R -> R) (lb ub : R), lb < ub -> (forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f y) -> (forall x : R, f lb <= x -> x <= f ub -> comp f g x = id x) -> (forall x : R, f lb <= x -> x <= f ub -> lb <= g x <= ub) -> forall x y : R, f lb <= x -> x < y -> y <= f ub -> g x < g yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_ok:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx, y:Rlb_le_x:f lb <= xx_lt_y:x < yy_le_ub:y <= f ubg x < g yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_ok:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx, y:Rlb_le_x:f lb <= xx_lt_y:x < yy_le_ub:y <= f ubx_encad:f lb <= x <= f ubg x < g yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_ok:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx, y:Rlb_le_x:f lb <= xx_lt_y:x < yy_le_ub:y <= f ubx_encad:f lb <= x <= f uby_encad:f lb <= y <= f ubg x < g yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_ok:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx, y:Rlb_le_x:f lb <= xx_lt_y:x < yy_le_ub:y <= f ubx_encad:f lb <= x <= f uby_encad:f lb <= y <= f ubgx_encad:lb <= g x <= ubg x < g yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_ok:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx, y:Rlb_le_x:f lb <= xx_lt_y:x < yy_le_ub:y <= f ubx_encad:f lb <= x <= f uby_encad:f lb <= y <= f ubgx_encad:lb <= g x <= ubgy_encad:lb <= g y <= ubg x < g yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_ok:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx, y:Rlb_le_x:f lb <= xx_lt_y:x < yy_le_ub:y <= f ubx_encad:f lb <= x <= f uby_encad:f lb <= y <= f ubgx_encad:lb <= g x <= ubgy_encad:lb <= g y <= ub~ g x < g y -> g x < g yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_ok:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx, y:Rlb_le_x:f lb <= xx_lt_y:x < yy_le_ub:y <= f ubx_encad:f lb <= x <= f uby_encad:f lb <= y <= f ubgx_encad:lb <= g x <= ubgy_encad:lb <= g y <= ubHfalse:~ g x < g yg x < g yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_ok:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx, y:Rlb_le_x:f lb <= xx_lt_y:x < yy_le_ub:y <= f ubx_encad:f lb <= x <= f uby_encad:f lb <= y <= f ubgx_encad:lb <= g x <= ubgy_encad:lb <= g y <= ubHfalse:~ g x < g yTemp:g y <= g xg x < g yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_ok:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx, y:Rlb_le_x:f lb <= xx_lt_y:x < yy_le_ub:y <= f ubx_encad:f lb <= x <= f uby_encad:f lb <= y <= f ubgx_encad:lb <= g x <= ubgy_encad:lb <= g y <= ubHfalse:~ g x < g yTemp:g y <= g xy <= xf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_ok:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx, y:Rlb_le_x:f lb <= xx_lt_y:x < yy_le_ub:y <= f ubx_encad:f lb <= x <= f uby_encad:f lb <= y <= f ubgx_encad:lb <= g x <= ubgy_encad:lb <= g y <= ubHfalse:~ g x < g yTemp:g y <= g xid y <= xf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_ok:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx, y:Rlb_le_x:f lb <= xx_lt_y:x < yy_le_ub:y <= f ubx_encad:f lb <= x <= f uby_encad:f lb <= y <= f ubgx_encad:lb <= g x <= ubgy_encad:lb <= g y <= ubHfalse:~ g x < g yTemp:g y <= g xid y <= id xf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_ok:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx, y:Rlb_le_x:f lb <= xx_lt_y:x < yy_le_ub:y <= f ubx_encad:f lb <= x <= f uby_encad:f lb <= y <= f ubgx_encad:lb <= g x <= ubgy_encad:lb <= g y <= ubHfalse:~ g x < g yTemp:g y <= g xcomp f g y <= id xf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_ok:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx, y:Rlb_le_x:f lb <= xx_lt_y:x < yy_le_ub:y <= f ubx_encad:f lb <= x <= f uby_encad:f lb <= y <= f ubgx_encad:lb <= g x <= ubgy_encad:lb <= g y <= ubHfalse:~ g x < g yTemp:g y <= g xcomp f g y <= comp f g xf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_ok:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx, y:Rlb_le_x:f lb <= xx_lt_y:x < yy_le_ub:y <= f ubx_encad:f lb <= x <= f uby_encad:f lb <= y <= f ubgx_encad:lb <= g x <= ubgy_encad:lb <= g y <= ubHfalse:~ g x < g yTemp:g y <= g xforall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 < ub -> f x0 <= f y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_ok:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx, y:Rlb_le_x:f lb <= xx_lt_y:x < yy_le_ub:y <= f ubx_encad:f lb <= x <= f uby_encad:f lb <= y <= f ubgx_encad:lb <= g x <= ubgy_encad:lb <= g y <= ubHfalse:~ g x < g yTemp:g y <= g xf_incr2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 < ub -> f x0 <= f y0comp f g y <= comp f g xf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_ok:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx, y:Rlb_le_x:f lb <= xx_lt_y:x < yy_le_ub:y <= f ubx_encad:f lb <= x <= f uby_encad:f lb <= y <= f ubgx_encad:lb <= g x <= ubgy_encad:lb <= g y <= ubHfalse:~ g x < g yTemp:g y <= g xforall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 < ub -> f x0 <= f y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_ok:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx, y:Rlb_le_x:f lb <= xx_lt_y:x < yy_le_ub:y <= f ubx_encad:f lb <= x <= f uby_encad:f lb <= y <= f ubgx_encad:lb <= g x <= ubgy_encad:lb <= g y <= ubHfalse:~ g x < g yTemp:g y <= g xm, n:Rlb_le_m:lb <= mm_le_n:m <= nn_lt_ub:n < ubf m <= f nf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_ok:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx, y:Rlb_le_x:f lb <= xx_lt_y:x < yy_le_ub:y <= f ubx_encad:f lb <= x <= f uby_encad:f lb <= y <= f ubgx_encad:lb <= g x <= ubgy_encad:lb <= g y <= ubHfalse:~ g x < g yTemp:g y <= g xm, n:Rlb_le_m:lb <= mm_le_n:m <= nn_lt_ub:n < ubm < n -> f m <= f nf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_ok:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx, y:Rlb_le_x:f lb <= xx_lt_y:x < yy_le_ub:y <= f ubx_encad:f lb <= x <= f uby_encad:f lb <= y <= f ubgx_encad:lb <= g x <= ubgy_encad:lb <= g y <= ubHfalse:~ g x < g yTemp:g y <= g xm, n:Rlb_le_m:lb <= mm_le_n:m <= nn_lt_ub:n < ubm = n -> f m <= f nintros; apply Rlt_le, f_incr, Rlt_le; assumption.f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_ok:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx, y:Rlb_le_x:f lb <= xx_lt_y:x < yy_le_ub:y <= f ubx_encad:f lb <= x <= f uby_encad:f lb <= y <= f ubgx_encad:lb <= g x <= ubgy_encad:lb <= g y <= ubHfalse:~ g x < g yTemp:g y <= g xm, n:Rlb_le_m:lb <= mm_le_n:m <= nn_lt_ub:n < ubm < n -> f m <= f nintros Hyp; rewrite Hyp; apply Req_le; reflexivity.f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_ok:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx, y:Rlb_le_x:f lb <= xx_lt_y:x < yy_le_ub:y <= f ubx_encad:f lb <= x <= f uby_encad:f lb <= y <= f ubgx_encad:lb <= g x <= ubgy_encad:lb <= g y <= ubHfalse:~ g x < g yTemp:g y <= g xm, n:Rlb_le_m:lb <= mm_le_n:m <= nn_lt_ub:n < ubm = n -> f m <= f nf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_ok:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx, y:Rlb_le_x:f lb <= xx_lt_y:x < yy_le_ub:y <= f ubx_encad:f lb <= x <= f uby_encad:f lb <= y <= f ubgx_encad:lb <= g x <= ubgy_encad:lb <= g y <= ubHfalse:~ g x < g yTemp:g y <= g xf_incr2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 < ub -> f x0 <= f y0comp f g y <= comp f g xf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_ok:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx, y:Rlb_le_x:f lb <= xx_lt_y:x < yy_le_ub:y <= f ubHfalse:g x < g y -> FalseTemp:g y <= g xf_incr2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 < ub -> f x0 <= f y0H:f lb <= xH0:x <= f ubH1:f lb <= yH2:y <= f ubH3:lb <= g xH4:g x <= ubH5:lb <= g yH6:g y <= ubg x < ubf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_ok:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx, y:Rlb_le_x:f lb <= xx_lt_y:x < yy_le_ub:y <= f ubHfalse:g x < g y -> FalseTemp:g y <= g xf_incr2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 < ub -> f x0 <= f y0H:f lb <= xH0:x <= f ubH1:f lb <= yH2:y <= f ubH3:lb <= g xH4:g x <= ubH5:lb <= g yH6:g y <= ubg x <> ubf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_ok:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx, y:Rlb_le_x:f lb <= xx_lt_y:x < yy_le_ub:y <= f ubHfalse:g x < g y -> FalseTemp:g y <= g xf_incr2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 < ub -> f x0 <= f y0H:f lb <= xH0:x <= f ubH1:f lb <= yH2:y <= f ubH3:lb <= g xH4:g x <= ubH5:lb <= g yH6:g y <= ubHf:g x = ubFalsef, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_ok:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx, y:Rlb_le_x:f lb <= xx_lt_y:x < yy_le_ub:y <= f ubHfalse:g x < g y -> FalseTemp:g y <= g xf_incr2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 < ub -> f x0 <= f y0H:f lb <= xH0:x <= f ubH1:f lb <= yH2:y <= f ubH3:lb <= g xH4:g x <= ubH5:lb <= g yH6:g y <= ubHf:g x = ubcomp f g x = f ubf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_ok:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx, y:Rlb_le_x:f lb <= xx_lt_y:x < yy_le_ub:y <= f ubHfalse:g x < g y -> FalseTemp:g y <= g xf_incr2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 < ub -> f x0 <= f y0H:f lb <= xH0:x <= f ubH1:f lb <= yH2:y <= f ubH3:lb <= g xH4:g x <= ubH5:lb <= g yH6:g y <= ubHf:g x = ubHtemp:comp f g x = f ubFalseunfold comp; rewrite Hf; reflexivity.f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_ok:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx, y:Rlb_le_x:f lb <= xx_lt_y:x < yy_le_ub:y <= f ubHfalse:g x < g y -> FalseTemp:g y <= g xf_incr2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 < ub -> f x0 <= f y0H:f lb <= xH0:x <= f ubH1:f lb <= yH2:y <= f ubH3:lb <= g xH4:g x <= ubH5:lb <= g yH6:g y <= ubHf:g x = ubcomp f g x = f ubf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_ok:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx, y:Rlb_le_x:f lb <= xx_lt_y:x < yy_le_ub:y <= f ubHfalse:g x < g y -> FalseTemp:g y <= g xf_incr2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 < ub -> f x0 <= f y0H:f lb <= xH0:x <= f ubH1:f lb <= yH2:y <= f ubH3:lb <= g xH4:g x <= ubH5:lb <= g yH6:g y <= ubHf:g x = ubHtemp:comp f g x = f ubFalsef, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_ok:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx, y:Rlb_le_x:f lb <= xx_lt_y:x < yy_le_ub:y <= f ubHfalse:g x < g y -> FalseTemp:g y <= g xf_incr2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 < ub -> f x0 <= f y0H:f lb <= xH0:x <= f ubH1:f lb <= yH2:y <= f ubH3:lb <= g xH4:g x <= ubH5:lb <= g yH6:g y <= ubHf:g x = ubHtemp:id x = f ubFalselra. Qed.f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_ok:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx, y:Rlb_le_x:f lb <= xx_lt_y:x < yy_le_ub:y <= f ubHfalse:g x < g y -> FalseTemp:g y <= g xf_incr2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 < ub -> f x0 <= f y0H:f lb <= xH0:x <= f ubH1:f lb <= yH2:y <= f ubH3:lb <= g xH4:g x <= ubH5:lb <= g yH6:g y <= ubHf:g x = ubHtemp:x = f ubFalseforall lb ub x : R, lb <= x <= ub -> derivable_pt id xforall lb ub x : R, lb <= x <= ub -> derivable_pt id xreg. Qed.lb, ub, x:RH:lb <= x <= ubderivable_pt id xforall (f g : R -> R) (lb ub x : R) (pr1 : derivable_pt f x) (pr2 : derivable_pt g x), lb < ub -> lb < x < ub -> (forall h : R, lb < h < ub -> f h = g h) -> derive_pt f x pr1 = derive_pt g x pr2forall (f g : R -> R) (lb ub x : R) (pr1 : derivable_pt f x) (pr2 : derivable_pt g x), lb < ub -> lb < x < ub -> (forall h : R, lb < h < ub -> f h = g h) -> derive_pt f x pr1 = derive_pt g x pr2f, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hderive_pt f x Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hforall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g ha, l:Ra_encad:lb < a < ubderivable_pt_abs f a l <-> derivable_pt_abs g a lf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g ha, l:Ra_encad:lb < a < ub(forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (a + h) - f a) / h - l) < eps) <-> (forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (a + h) - g a) / h - l) < eps)f, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g ha, l:Ra_encad:lb < a < ub(forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (a + h) - f a) / h - l) < eps) -> forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (a + h) - g a) / h - l) < epsf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g ha, l:Ra_encad:lb < a < ub(forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (a + h) - g a) / h - l) < eps) -> forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (a + h) - f a) / h - l) < epsf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g ha, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (a + h) - f a) / h - l) < eps0eps:Reps_pos:0 < epsexists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (a + h) - g a) / h - l) < epsf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g ha, l:Ra_encad:lb < a < ub(forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (a + h) - g a) / h - l) < eps) -> forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (a + h) - f a) / h - l) < epsf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g ha, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((f (a + h) - f a) / h - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (a + h) - f a) / h - l) < epsexists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((g (a + h) - g a) / h - l) < epsf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g ha, l:Ra_encad:lb < a < ub(forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (a + h) - g a) / h - l) < eps) -> forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (a + h) - f a) / h - l) < epsf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g ha, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((f (a + h) - f a) / h - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (a + h) - f a) / h - l) < epsRmin delta (Rmin (ub - a) (a - lb)) > 0f, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g ha, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((f (a + h) - f a) / h - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (a + h) - f a) / h - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((g (a + h) - g a) / h - l) < epsf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g ha, l:Ra_encad:lb < a < ub(forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (a + h) - g a) / h - l) < eps) -> forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (a + h) - f a) / h - l) < epsf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prglb, ub, a:Ra_encad:lb < a < ubdelta:posrealRmin delta (Rmin (ub - a) (a - lb)) > 0f, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g ha, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((f (a + h) - f a) / h - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (a + h) - f a) / h - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((g (a + h) - g a) / h - l) < epsf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g ha, l:Ra_encad:lb < a < ub(forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (a + h) - g a) / h - l) < eps) -> forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (a + h) - f a) / h - l) < epsf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g ha, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((f (a + h) - f a) / h - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (a + h) - f a) / h - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((g (a + h) - g a) / h - l) < epsf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g ha, l:Ra_encad:lb < a < ub(forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (a + h) - g a) / h - l) < eps) -> forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (a + h) - f a) / h - l) < epsf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g ha, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((f (a + h) - f a) / h - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (a + h) - f a) / h - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0forall h : R, h <> 0 -> Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |} -> Rabs ((g (a + h) - g a) / h - l) < epsf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g ha, l:Ra_encad:lb < a < ub(forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (a + h) - g a) / h - l) < eps) -> forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (a + h) - f a) / h - l) < epsf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (a + h0) - f a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}Rabs ((g (a + h) - g a) / h - l) < epsf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g ha, l:Ra_encad:lb < a < ub(forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (a + h) - g a) / h - l) < eps) -> forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (a + h) - f a) / h - l) < epsf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (a + h0) - f a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}Rabs ((f (a + h) - f a) / h - l) < epsf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (a + h0) - f a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}f (a + h) - f a = g (a + h) - g af, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g ha, l:Ra_encad:lb < a < ub(forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (a + h) - g a) / h - l) < eps) -> forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (a + h) - f a) / h - l) < epsf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:RHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, (h0 = 0 -> False) -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, (h0 = 0 -> False) -> Rabs h0 < delta -> Rabs ((f (a + h0) - f a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h = 0 -> Falseh_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}H:lb < xH0:x < ubH1:lb < aH2:a < ubRabs h < deltaf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (a + h0) - f a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}f (a + h) - f a = g (a + h) - g af, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g ha, l:Ra_encad:lb < a < ub(forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (a + h) - g a) / h - l) < eps) -> forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (a + h) - f a) / h - l) < epsf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:RHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, (h0 = 0 -> False) -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, (h0 = 0 -> False) -> Rabs h0 < delta -> Rabs ((f (a + h0) - f a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h = 0 -> Falseh_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}H:lb < xH0:x < ubH1:lb < aH2:a < ubRabs h < Rmin delta (Rmin (ub - a) (a - lb))f, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:RHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, (h0 = 0 -> False) -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, (h0 = 0 -> False) -> Rabs h0 < delta -> Rabs ((f (a + h0) - f a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h = 0 -> Falseh_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}H:lb < xH0:x < ubH1:lb < aH2:a < ubRmin delta (Rmin (ub - a) (a - lb)) <= deltaf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (a + h0) - f a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}f (a + h) - f a = g (a + h) - g af, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g ha, l:Ra_encad:lb < a < ub(forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (a + h) - g a) / h - l) < eps) -> forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (a + h) - f a) / h - l) < epsf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:RHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, (h0 = 0 -> False) -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, (h0 = 0 -> False) -> Rabs h0 < delta -> Rabs ((f (a + h0) - f a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h = 0 -> Falseh_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}H:lb < xH0:x < ubH1:lb < aH2:a < ubRmin delta (Rmin (ub - a) (a - lb)) <= deltaf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (a + h0) - f a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}f (a + h) - f a = g (a + h) - g af, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g ha, l:Ra_encad:lb < a < ub(forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (a + h) - g a) / h - l) < eps) -> forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (a + h) - f a) / h - l) < epsf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (a + h0) - f a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}f (a + h) - f a = g (a + h) - g af, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g ha, l:Ra_encad:lb < a < ub(forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (a + h) - g a) / h - l) < eps) -> forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (a + h) - f a) / h - l) < epsf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (a + h0) - f a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}forall h0 : R, lb < h0 < ub -> - f h0 = - g h0f, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (a + h0) - f a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0f (a + h) - f a = g (a + h) - g af, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g ha, l:Ra_encad:lb < a < ub(forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (a + h) - g a) / h - l) < eps) -> forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (a + h) - f a) / h - l) < epsf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (a + h0) - f a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0f (a + h) - f a = g (a + h) - g af, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g ha, l:Ra_encad:lb < a < ub(forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (a + h) - g a) / h - l) < eps) -> forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (a + h) - f a) / h - l) < epsf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (a + h0) - f a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0g (a + h) + - f a = g (a + h) + - g af, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (a + h0) - f a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0lb < a + h < ubf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g ha, l:Ra_encad:lb < a < ub(forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (a + h) - g a) / h - l) < eps) -> forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (a + h) - f a) / h - l) < epsf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (a + h0) - f a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0g (a + h) + - g a = g (a + h) + - g af, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (a + h0) - f a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0lb < a < ubf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (a + h0) - f a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0lb < a + h < ubf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g ha, l:Ra_encad:lb < a < ub(forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (a + h) - g a) / h - l) < eps) -> forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (a + h) - f a) / h - l) < epsf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (a + h0) - f a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0lb < a < ubf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (a + h0) - f a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0lb < a + h < ubf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g ha, l:Ra_encad:lb < a < ub(forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (a + h) - g a) / h - l) < eps) -> forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (a + h) - f a) / h - l) < epsf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (a + h0) - f a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0lb < a + h < ubf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g ha, l:Ra_encad:lb < a < ub(forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (a + h) - g a) / h - l) < eps) -> forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (a + h) - f a) / h - l) < epsf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (a + h0) - f a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0forall x0 y : R, Rabs x0 < Rabs y -> y > 0 -> x0 < yf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (a + h0) - f a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0Sublemma2:forall x0 y : R, Rabs x0 < Rabs y -> y > 0 -> x0 < ylb < a + h < ubf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g ha, l:Ra_encad:lb < a < ub(forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (a + h) - g a) / h - l) < eps) -> forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (a + h) - f a) / h - l) < epsf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (a + h0) - f a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0m, n:RHyp_abs:Rabs m < Rabs ny_pos:n > 0m < nf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (a + h0) - f a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0Sublemma2:forall x0 y : R, Rabs x0 < Rabs y -> y > 0 -> x0 < ylb < a + h < ubf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g ha, l:Ra_encad:lb < a < ub(forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (a + h) - g a) / h - l) < eps) -> forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (a + h) - f a) / h - l) < epsf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (a + h0) - f a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0m, n:RHyp_abs:Rabs m < Rabs ny_pos:n > 0m < Rabs nf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (a + h0) - f a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0m, n:RHyp_abs:Rabs m < Rabs ny_pos:n > 0Rabs n <= nf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (a + h0) - f a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0Sublemma2:forall x0 y : R, Rabs x0 < Rabs y -> y > 0 -> x0 < ylb < a + h < ubf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g ha, l:Ra_encad:lb < a < ub(forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (a + h) - g a) / h - l) < eps) -> forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (a + h) - f a) / h - l) < epsf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (a + h0) - f a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0m, n:RHyp_abs:Rabs m < Rabs ny_pos:n > 0Rabs n <= nf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (a + h0) - f a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0Sublemma2:forall x0 y : R, Rabs x0 < Rabs y -> y > 0 -> x0 < ylb < a + h < ubf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g ha, l:Ra_encad:lb < a < ub(forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (a + h) - g a) / h - l) < eps) -> forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (a + h) - f a) / h - l) < epsf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (a + h0) - f a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0Sublemma2:forall x0 y : R, Rabs x0 < Rabs y -> y > 0 -> x0 < ylb < a + h < ubf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g ha, l:Ra_encad:lb < a < ub(forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (a + h) - g a) / h - l) < eps) -> forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (a + h) - f a) / h - l) < epsf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (a + h0) - f a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0Sublemma2:forall x0 y : R, Rabs x0 < Rabs y -> y > 0 -> x0 < ylb < a + hf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (a + h0) - f a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0Sublemma2:forall x0 y : R, Rabs x0 < Rabs y -> y > 0 -> x0 < ya + h < ubf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g ha, l:Ra_encad:lb < a < ub(forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (a + h) - g a) / h - l) < eps) -> forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (a + h) - f a) / h - l) < epsf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (a + h0) - f a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0Sublemma2:forall x0 y : R, Rabs x0 < Rabs y -> y > 0 -> x0 < yforall x0 y z : R, - z < y - x0 -> x0 < y + zf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (a + h0) - f a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0Sublemma2:forall x0 y : R, Rabs x0 < Rabs y -> y > 0 -> x0 < ySublemma:forall x0 y z : R, - z < y - x0 -> x0 < y + zlb < a + hf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (a + h0) - f a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0Sublemma2:forall x0 y : R, Rabs x0 < Rabs y -> y > 0 -> x0 < ya + h < ubf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g ha, l:Ra_encad:lb < a < ub(forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (a + h) - g a) / h - l) < eps) -> forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (a + h) - f a) / h - l) < epsf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (a + h0) - f a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0Sublemma2:forall x0 y : R, Rabs x0 < Rabs y -> y > 0 -> x0 < ySublemma:forall x0 y z : R, - z < y - x0 -> x0 < y + zlb < a + hf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (a + h0) - f a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0Sublemma2:forall x0 y : R, Rabs x0 < Rabs y -> y > 0 -> x0 < ya + h < ubf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g ha, l:Ra_encad:lb < a < ub(forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (a + h) - g a) / h - l) < eps) -> forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (a + h) - f a) / h - l) < epsf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (a + h0) - f a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0Sublemma2:forall x0 y : R, Rabs x0 < Rabs y -> y > 0 -> x0 < ySublemma:forall x0 y z : R, - z < y - x0 -> x0 < y + z- h < a - lbf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (a + h0) - f a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0Sublemma2:forall x0 y : R, Rabs x0 < Rabs y -> y > 0 -> x0 < ya + h < ubf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g ha, l:Ra_encad:lb < a < ub(forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (a + h) - g a) / h - l) < eps) -> forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (a + h) - f a) / h - l) < epsf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (a + h0) - f a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0Sublemma2:forall x0 y : R, Rabs x0 < Rabs y -> y > 0 -> x0 < ySublemma:forall x0 y z : R, - z < y - x0 -> x0 < y + zRabs (- h) < Rabs (a - lb)f, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (a + h0) - f a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0Sublemma2:forall x0 y : R, Rabs x0 < Rabs y -> y > 0 -> x0 < ySublemma:forall x0 y z : R, - z < y - x0 -> x0 < y + za - lb > 0f, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (a + h0) - f a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0Sublemma2:forall x0 y : R, Rabs x0 < Rabs y -> y > 0 -> x0 < ya + h < ubf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g ha, l:Ra_encad:lb < a < ub(forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (a + h) - g a) / h - l) < eps) -> forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (a + h) - f a) / h - l) < epsf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (a + h0) - f a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0Sublemma2:forall x0 y : R, Rabs x0 < Rabs y -> y > 0 -> x0 < ySublemma:forall x0 y z : R, - z < y - x0 -> x0 < y + zRabs h < Rabs (a - lb)f, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (a + h0) - f a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0Sublemma2:forall x0 y : R, Rabs x0 < Rabs y -> y > 0 -> x0 < ySublemma:forall x0 y z : R, - z < y - x0 -> x0 < y + za - lb > 0f, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (a + h0) - f a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0Sublemma2:forall x0 y : R, Rabs x0 < Rabs y -> y > 0 -> x0 < ya + h < ubf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g ha, l:Ra_encad:lb < a < ub(forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (a + h) - g a) / h - l) < eps) -> forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (a + h) - f a) / h - l) < epsf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (a + h0) - f a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0Sublemma2:forall x0 y : R, Rabs x0 < Rabs y -> y > 0 -> x0 < ySublemma:forall x0 y z : R, - z < y - x0 -> x0 < y + za - lb > 0f, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (a + h0) - f a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0Sublemma2:forall x0 y : R, Rabs x0 < Rabs y -> y > 0 -> x0 < ya + h < ubf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g ha, l:Ra_encad:lb < a < ub(forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (a + h) - g a) / h - l) < eps) -> forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (a + h) - f a) / h - l) < epsf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (a + h0) - f a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0Sublemma2:forall x0 y : R, Rabs x0 < Rabs y -> y > 0 -> x0 < ya + h < ubf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g ha, l:Ra_encad:lb < a < ub(forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (a + h) - g a) / h - l) < eps) -> forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (a + h) - f a) / h - l) < epsf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (a + h0) - f a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0Sublemma2:forall x0 y : R, Rabs x0 < Rabs y -> y > 0 -> x0 < yforall x0 y z : R, y < z - x0 -> x0 + y < zf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (a + h0) - f a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0Sublemma2:forall x0 y : R, Rabs x0 < Rabs y -> y > 0 -> x0 < ySublemma:forall x0 y z : R, y < z - x0 -> x0 + y < za + h < ubf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g ha, l:Ra_encad:lb < a < ub(forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (a + h) - g a) / h - l) < eps) -> forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (a + h) - f a) / h - l) < epsf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (a + h0) - f a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0Sublemma2:forall x0 y : R, Rabs x0 < Rabs y -> y > 0 -> x0 < ySublemma:forall x0 y z : R, y < z - x0 -> x0 + y < za + h < ubf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g ha, l:Ra_encad:lb < a < ub(forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (a + h) - g a) / h - l) < eps) -> forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (a + h) - f a) / h - l) < epsf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (a + h0) - f a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0Sublemma2:forall x0 y : R, Rabs x0 < Rabs y -> y > 0 -> x0 < ySublemma:forall x0 y z : R, y < z - x0 -> x0 + y < zh < ub - af, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g ha, l:Ra_encad:lb < a < ub(forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (a + h) - g a) / h - l) < eps) -> forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (a + h) - f a) / h - l) < epsf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (a + h0) - f a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0Sublemma2:forall x0 y : R, Rabs x0 < Rabs y -> y > 0 -> x0 < ySublemma:forall x0 y z : R, y < z - x0 -> x0 + y < zRabs h < Rabs (ub - a)f, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (a + h0) - f a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0Sublemma2:forall x0 y : R, Rabs x0 < Rabs y -> y > 0 -> x0 < ySublemma:forall x0 y z : R, y < z - x0 -> x0 + y < zub - a > 0f, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g ha, l:Ra_encad:lb < a < ub(forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (a + h) - g a) / h - l) < eps) -> forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (a + h) - f a) / h - l) < epsf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (a + h0) - f a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0Sublemma2:forall x0 y : R, Rabs x0 < Rabs y -> y > 0 -> x0 < ySublemma:forall x0 y z : R, y < z - x0 -> x0 + y < zub - a > 0f, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g ha, l:Ra_encad:lb < a < ub(forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (a + h) - g a) / h - l) < eps) -> forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (a + h) - f a) / h - l) < epsf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g ha, l:Ra_encad:lb < a < ub(forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (a + h) - g a) / h - l) < eps) -> forall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (a + h) - f a) / h - l) < epsf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g ha, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (a + h) - g a) / h - l) < eps0eps:Reps_pos:0 < epsexists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (a + h) - f a) / h - l) < epsf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g ha, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((g (a + h) - g a) / h - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (a + h) - g a) / h - l) < epsexists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((f (a + h) - f a) / h - l) < epsf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g ha, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((g (a + h) - g a) / h - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (a + h) - g a) / h - l) < epsRmin delta (Rmin (ub - a) (a - lb)) > 0f, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g ha, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((g (a + h) - g a) / h - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (a + h) - g a) / h - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((f (a + h) - f a) / h - l) < epsf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prglb, ub, a:Ra_encad:lb < a < ubdelta:posrealRmin delta (Rmin (ub - a) (a - lb)) > 0f, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g ha, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((g (a + h) - g a) / h - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (a + h) - g a) / h - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((f (a + h) - f a) / h - l) < epsf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g ha, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((g (a + h) - g a) / h - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (a + h) - g a) / h - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((f (a + h) - f a) / h - l) < epsf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g ha, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((g (a + h) - g a) / h - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (a + h) - g a) / h - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0forall h : R, h <> 0 -> Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |} -> Rabs ((f (a + h) - f a) / h - l) < epsf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((g (a + h0) - g a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((g (a + h0) - g a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}Rabs ((f (a + h) - f a) / h - l) < epsf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((g (a + h0) - g a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((g (a + h0) - g a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}Rabs ((g (a + h) - g a) / h - l) < epsf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((g (a + h0) - g a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((g (a + h0) - g a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}g (a + h) - g a = f (a + h) - f af, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:RHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, (h0 = 0 -> False) -> Rabs h0 < delta0 -> Rabs ((g (a + h0) - g a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, (h0 = 0 -> False) -> Rabs h0 < delta -> Rabs ((g (a + h0) - g a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h = 0 -> Falseh_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}H:lb < xH0:x < ubH1:lb < aH2:a < ubRabs h < deltaf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((g (a + h0) - g a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((g (a + h0) - g a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}g (a + h) - g a = f (a + h) - f af, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:RHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, (h0 = 0 -> False) -> Rabs h0 < delta0 -> Rabs ((g (a + h0) - g a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, (h0 = 0 -> False) -> Rabs h0 < delta -> Rabs ((g (a + h0) - g a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h = 0 -> Falseh_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}H:lb < xH0:x < ubH1:lb < aH2:a < ubRabs h < Rmin delta (Rmin (ub - a) (a - lb))f, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:RHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, (h0 = 0 -> False) -> Rabs h0 < delta0 -> Rabs ((g (a + h0) - g a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, (h0 = 0 -> False) -> Rabs h0 < delta -> Rabs ((g (a + h0) - g a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h = 0 -> Falseh_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}H:lb < xH0:x < ubH1:lb < aH2:a < ubRmin delta (Rmin (ub - a) (a - lb)) <= deltaf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((g (a + h0) - g a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((g (a + h0) - g a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}g (a + h) - g a = f (a + h) - f af, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:RHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, (h0 = 0 -> False) -> Rabs h0 < delta0 -> Rabs ((g (a + h0) - g a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, (h0 = 0 -> False) -> Rabs h0 < delta -> Rabs ((g (a + h0) - g a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h = 0 -> Falseh_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}H:lb < xH0:x < ubH1:lb < aH2:a < ubRmin delta (Rmin (ub - a) (a - lb)) <= deltaf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((g (a + h0) - g a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((g (a + h0) - g a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}g (a + h) - g a = f (a + h) - f af, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((g (a + h0) - g a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((g (a + h0) - g a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}g (a + h) - g a = f (a + h) - f af, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((g (a + h0) - g a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((g (a + h0) - g a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}forall h0 : R, lb < h0 < ub -> - f h0 = - g h0f, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((g (a + h0) - g a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((g (a + h0) - g a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0g (a + h) - g a = f (a + h) - f af, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((g (a + h0) - g a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((g (a + h0) - g a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0g (a + h) - g a = f (a + h) - f af, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((g (a + h0) - g a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((g (a + h0) - g a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0g (a + h) + - g a = g (a + h) + - f af, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((g (a + h0) - g a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((g (a + h0) - g a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0lb < a + h < ubf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((g (a + h0) - g a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((g (a + h0) - g a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0g (a + h) + - g a = g (a + h) + - g af, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((g (a + h0) - g a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((g (a + h0) - g a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0lb < a < ubf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((g (a + h0) - g a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((g (a + h0) - g a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0lb < a + h < ubf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((g (a + h0) - g a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((g (a + h0) - g a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0lb < a < ubf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((g (a + h0) - g a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((g (a + h0) - g a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0lb < a + h < ubf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((g (a + h0) - g a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((g (a + h0) - g a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0lb < a + h < ubf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((g (a + h0) - g a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((g (a + h0) - g a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0forall x0 y : R, Rabs x0 < Rabs y -> y > 0 -> x0 < yf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((g (a + h0) - g a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((g (a + h0) - g a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0Sublemma2:forall x0 y : R, Rabs x0 < Rabs y -> y > 0 -> x0 < ylb < a + h < ubf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((g (a + h0) - g a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((g (a + h0) - g a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0m, n:RHyp_abs:Rabs m < Rabs ny_pos:n > 0m < nf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((g (a + h0) - g a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((g (a + h0) - g a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0Sublemma2:forall x0 y : R, Rabs x0 < Rabs y -> y > 0 -> x0 < ylb < a + h < ubf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((g (a + h0) - g a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((g (a + h0) - g a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0m, n:RHyp_abs:Rabs m < Rabs ny_pos:n > 0m < Rabs nf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((g (a + h0) - g a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((g (a + h0) - g a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0m, n:RHyp_abs:Rabs m < Rabs ny_pos:n > 0Rabs n <= nf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((g (a + h0) - g a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((g (a + h0) - g a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0Sublemma2:forall x0 y : R, Rabs x0 < Rabs y -> y > 0 -> x0 < ylb < a + h < ubf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((g (a + h0) - g a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((g (a + h0) - g a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0m, n:RHyp_abs:Rabs m < Rabs ny_pos:n > 0Rabs n <= nf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((g (a + h0) - g a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((g (a + h0) - g a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0Sublemma2:forall x0 y : R, Rabs x0 < Rabs y -> y > 0 -> x0 < ylb < a + h < ubf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((g (a + h0) - g a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((g (a + h0) - g a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0Sublemma2:forall x0 y : R, Rabs x0 < Rabs y -> y > 0 -> x0 < ylb < a + h < ubf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((g (a + h0) - g a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((g (a + h0) - g a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0Sublemma2:forall x0 y : R, Rabs x0 < Rabs y -> y > 0 -> x0 < ylb < a + hf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((g (a + h0) - g a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((g (a + h0) - g a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0Sublemma2:forall x0 y : R, Rabs x0 < Rabs y -> y > 0 -> x0 < ya + h < ubf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((g (a + h0) - g a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((g (a + h0) - g a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0Sublemma2:forall x0 y : R, Rabs x0 < Rabs y -> y > 0 -> x0 < yforall x0 y z : R, - z < y - x0 -> x0 < y + zf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((g (a + h0) - g a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((g (a + h0) - g a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0Sublemma2:forall x0 y : R, Rabs x0 < Rabs y -> y > 0 -> x0 < ySublemma:forall x0 y z : R, - z < y - x0 -> x0 < y + zlb < a + hf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((g (a + h0) - g a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((g (a + h0) - g a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0Sublemma2:forall x0 y : R, Rabs x0 < Rabs y -> y > 0 -> x0 < ya + h < ubf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((g (a + h0) - g a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((g (a + h0) - g a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0Sublemma2:forall x0 y : R, Rabs x0 < Rabs y -> y > 0 -> x0 < ySublemma:forall x0 y z : R, - z < y - x0 -> x0 < y + zlb < a + hf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((g (a + h0) - g a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((g (a + h0) - g a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0Sublemma2:forall x0 y : R, Rabs x0 < Rabs y -> y > 0 -> x0 < ya + h < ubf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((g (a + h0) - g a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((g (a + h0) - g a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0Sublemma2:forall x0 y : R, Rabs x0 < Rabs y -> y > 0 -> x0 < ySublemma:forall x0 y z : R, - z < y - x0 -> x0 < y + z- h < a - lbf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((g (a + h0) - g a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((g (a + h0) - g a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0Sublemma2:forall x0 y : R, Rabs x0 < Rabs y -> y > 0 -> x0 < ya + h < ubf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((g (a + h0) - g a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((g (a + h0) - g a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0Sublemma2:forall x0 y : R, Rabs x0 < Rabs y -> y > 0 -> x0 < ySublemma:forall x0 y z : R, - z < y - x0 -> x0 < y + zRabs (- h) < Rabs (a - lb)f, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((g (a + h0) - g a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((g (a + h0) - g a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0Sublemma2:forall x0 y : R, Rabs x0 < Rabs y -> y > 0 -> x0 < ySublemma:forall x0 y z : R, - z < y - x0 -> x0 < y + za - lb > 0f, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((g (a + h0) - g a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((g (a + h0) - g a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0Sublemma2:forall x0 y : R, Rabs x0 < Rabs y -> y > 0 -> x0 < ya + h < ubf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((g (a + h0) - g a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((g (a + h0) - g a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0Sublemma2:forall x0 y : R, Rabs x0 < Rabs y -> y > 0 -> x0 < ySublemma:forall x0 y z : R, - z < y - x0 -> x0 < y + zRabs h < Rabs (a - lb)f, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((g (a + h0) - g a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((g (a + h0) - g a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0Sublemma2:forall x0 y : R, Rabs x0 < Rabs y -> y > 0 -> x0 < ySublemma:forall x0 y z : R, - z < y - x0 -> x0 < y + za - lb > 0f, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((g (a + h0) - g a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((g (a + h0) - g a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0Sublemma2:forall x0 y : R, Rabs x0 < Rabs y -> y > 0 -> x0 < ya + h < ubf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((g (a + h0) - g a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((g (a + h0) - g a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0Sublemma2:forall x0 y : R, Rabs x0 < Rabs y -> y > 0 -> x0 < ySublemma:forall x0 y z : R, - z < y - x0 -> x0 < y + za - lb > 0f, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((g (a + h0) - g a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((g (a + h0) - g a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0Sublemma2:forall x0 y : R, Rabs x0 < Rabs y -> y > 0 -> x0 < ya + h < ubf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((g (a + h0) - g a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((g (a + h0) - g a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0Sublemma2:forall x0 y : R, Rabs x0 < Rabs y -> y > 0 -> x0 < ya + h < ubf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((g (a + h0) - g a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((g (a + h0) - g a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0Sublemma2:forall x0 y : R, Rabs x0 < Rabs y -> y > 0 -> x0 < yforall x0 y z : R, y < z - x0 -> x0 + y < zf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((g (a + h0) - g a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((g (a + h0) - g a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0Sublemma2:forall x0 y : R, Rabs x0 < Rabs y -> y > 0 -> x0 < ySublemma:forall x0 y z : R, y < z - x0 -> x0 + y < za + h < ubf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((g (a + h0) - g a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((g (a + h0) - g a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0Sublemma2:forall x0 y : R, Rabs x0 < Rabs y -> y > 0 -> x0 < ySublemma:forall x0 y z : R, y < z - x0 -> x0 + y < za + h < ubf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((g (a + h0) - g a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((g (a + h0) - g a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0Sublemma2:forall x0 y : R, Rabs x0 < Rabs y -> y > 0 -> x0 < ySublemma:forall x0 y z : R, y < z - x0 -> x0 + y < zh < ub - af, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((g (a + h0) - g a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((g (a + h0) - g a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0Sublemma2:forall x0 y : R, Rabs x0 < Rabs y -> y > 0 -> x0 < ySublemma:forall x0 y z : R, y < z - x0 -> x0 + y < zRabs h < Rabs (ub - a)f, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((g (a + h0) - g a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((g (a + h0) - g a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0Sublemma2:forall x0 y : R, Rabs x0 < Rabs y -> y > 0 -> x0 < ySublemma:forall x0 y z : R, y < z - x0 -> x0 + y < zub - a > 0f, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0a, l:Ra_encad:lb < a < ubHyp:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((g (a + h0) - g a) / h0 - l) < eps0eps:Reps_pos:0 < epsdelta:posrealHyp2:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((g (a + h0) - g a) / h0 - l) < epsPos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0h:Rh_neq:h <> 0h_encad:Rabs h < {| pos := Rmin delta (Rmin (ub - a) (a - lb)); cond_pos := Pos_cond |}local_eq2:forall h0 : R, lb < h0 < ub -> - f h0 = - g h0Sublemma2:forall x0 y : R, Rabs x0 < Rabs y -> y > 0 -> x0 < ySublemma:forall x0 y z : R, y < z - x0 -> x0 + y < zub - a > 0f, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f xPrg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:{l : R | derivable_pt_abs f x l}Prg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:{l : R | derivable_pt_abs f x l}Prg:{l : R | derivable_pt_abs g x l}lb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x0 l : R, lb < x0 < ub -> derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 lderive_pt f x Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:{l : R | derivable_pt_abs f x l}Prg:{l : R | derivable_pt_abs g x l}lb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x1 l : R, lb < x1 < ub -> derivable_pt_abs f x1 l <-> derivable_pt_abs g x1 lx0:Rp:derivable_pt_abs f x x0derive_pt f x (exist (fun l : R => derivable_pt_abs f x l) x0 p) = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:{l : R | derivable_pt_abs f x l}Prg:{l : R | derivable_pt_abs g x l}lb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x2 l : R, lb < x2 < ub -> derivable_pt_abs f x2 l <-> derivable_pt_abs g x2 lx0:Rp:derivable_pt_abs f x x0x1:Rp0:derivable_pt_abs g x x1derive_pt f x (exist (fun l : R => derivable_pt_abs f x l) x0 p) = derive_pt g x (exist (fun l : R => derivable_pt_abs g x l) x1 p0)f, g:R -> Rlb, ub, x:RPrf:{l : R | derivable_pt_abs f x l}Prg:{l : R | derivable_pt_abs g x l}lb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x2 l : R, lb < x2 < ub -> derivable_pt_abs f x2 l <-> derivable_pt_abs g x2 lx0:Rp:derivable_pt_abs f x x0x1:Rp0:derivable_pt_abs g x x1Temp:derivable_pt_abs g x x0derive_pt f x (exist (fun l : R => derivable_pt_abs f x l) x0 p) = derive_pt g x (exist (fun l : R => derivable_pt_abs g x l) x1 p0)f, g:R -> Rlb, ub, x:RPrf:{l : R | derivable_pt_abs f x l}Prg:{l : R | derivable_pt_abs g x l}lb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x2 l : R, lb < x2 < ub -> derivable_pt_abs f x2 l <-> derivable_pt_abs g x2 lx0:Rp:derivable_pt_abs f x x0x1:Rp0:derivable_pt_abs g x x1Temp:derivable_pt_abs f x x0lb < x < ubf, g:R -> Rlb, ub, x:RPrf:{l : R | derivable_pt_abs f x l}Prg:{l : R | derivable_pt_abs g x l}lb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x2 l : R, lb < x2 < ub -> derivable_pt_abs f x2 l <-> derivable_pt_abs g x2 lx0:Rp:derivable_pt_lim f x x0x1:Rp0:derivable_pt_abs g x x1Temp:derivable_pt_abs g x x0derive_pt f x (exist (fun l : R => derivable_pt_abs f x l) x0 p) = derive_pt g x (exist (fun l : R => derivable_pt_abs g x l) x1 p0)f, g:R -> Rlb, ub, x:RPrf:{l : R | derivable_pt_abs f x l}Prg:{l : R | derivable_pt_abs g x l}lb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x2 l : R, lb < x2 < ub -> derivable_pt_abs f x2 l <-> derivable_pt_abs g x2 lx0:Rp:derivable_pt_abs f x x0x1:Rp0:derivable_pt_abs g x x1Temp:derivable_pt_abs f x x0lb < x < ubf, g:R -> Rlb, ub, x:RPrf:{l : R | derivable_pt_abs f x l}Prg:{l : R | derivable_pt_abs g x l}lb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x2 l : R, lb < x2 < ub -> derivable_pt_abs f x2 l <-> derivable_pt_abs g x2 lx0:Rp:derivable_pt_lim f x x0x1:Rp0:derivable_pt_lim g x x1Temp:derivable_pt_abs g x x0derive_pt f x (exist (fun l : R => derivable_pt_abs f x l) x0 p) = derive_pt g x (exist (fun l : R => derivable_pt_abs g x l) x1 p0)f, g:R -> Rlb, ub, x:RPrf:{l : R | derivable_pt_abs f x l}Prg:{l : R | derivable_pt_abs g x l}lb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x2 l : R, lb < x2 < ub -> derivable_pt_abs f x2 l <-> derivable_pt_abs g x2 lx0:Rp:derivable_pt_abs f x x0x1:Rp0:derivable_pt_abs g x x1Temp:derivable_pt_abs f x x0lb < x < ubf, g:R -> Rlb, ub, x:RPrf:{l : R | derivable_pt_abs f x l}Prg:{l : R | derivable_pt_abs g x l}lb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x2 l : R, lb < x2 < ub -> derivable_pt_abs f x2 l <-> derivable_pt_abs g x2 lx0:Rp:derivable_pt_lim f x x0x1:Rp0:derivable_pt_lim g x x1Temp:derivable_pt_abs g x x0x0 = x1f, g:R -> Rlb, ub, x:RPrf:{l : R | derivable_pt_abs f x l}Prg:{l : R | derivable_pt_abs g x l}lb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x2 l : R, lb < x2 < ub -> derivable_pt_abs f x2 l <-> derivable_pt_abs g x2 lx0:Rp:derivable_pt_abs f x x0x1:Rp0:derivable_pt_abs g x x1Temp:derivable_pt_abs f x x0lb < x < ubassumption. Qed. (* begin hide *)f, g:R -> Rlb, ub, x:RPrf:{l : R | derivable_pt_abs f x l}Prg:{l : R | derivable_pt_abs g x l}lb_lt_ub:lb < ubx_encad:lb < x < ublocal_eq:forall h : R, lb < h < ub -> f h = g hH:forall x2 l : R, lb < x2 < ub -> derivable_pt_abs f x2 l <-> derivable_pt_abs g x2 lx0:Rp:derivable_pt_abs f x x0x1:Rp0:derivable_pt_abs g x x1Temp:derivable_pt_abs f x x0lb < x < ubforall f g : R -> R, (forall x y : R, x < y -> f x < f y) -> (forall x : R, comp f g x = id x) -> forall x : R, comp g f x = id xforall f g : R -> R, (forall x y : R, x < y -> f x < f y) -> (forall x : R, comp f g x = id x) -> forall x : R, comp g f x = id xf, g:R -> Rf_incr:forall x0 y : R, x0 < y -> f x0 < f yHyp:forall x0 : R, comp f g x0 = id x0x:Rcomp g f x = id xf, g:R -> Rf_incr:forall x0 y : R, x0 < y -> f x0 < f yHyp:forall x0 : R, comp f g x0 = id x0x:Rforall x0 : R, f (g (f x0)) = f x0f, g:R -> Rf_incr:forall x0 y : R, x0 < y -> f x0 < f yHyp:forall x0 : R, comp f g x0 = id x0x:RH:forall x0 : R, f (g (f x0)) = f x0comp g f x = id xf, g:R -> Rf_incr:forall x0 y : R, x0 < y -> f x0 < f yHyp:forall x0 : R, comp f g x0 = id x0x:RH:forall x0 : R, f (g (f x0)) = f x0comp g f x = id xf, g:R -> Rf_incr:forall x0 y : R, x0 < y -> f x0 < f yHyp:forall x0 : R, comp f g x0 = id x0x:RH:forall x0 : R, f (g (f x0)) = f x0forall x0 y : R, f x0 = f y -> x0 = yf, g:R -> Rf_incr:forall x0 y : R, x0 < y -> f x0 < f yHyp:forall x0 : R, comp f g x0 = id x0x:RH:forall x0 : R, f (g (f x0)) = f x0f_inj:forall x0 y : R, f x0 = f y -> x0 = ycomp g f x = id xf, g:R -> Rf_incr:forall x0 y : R, x0 < y -> f x0 < f yHyp:forall x0 : R, comp f g x0 = id x0x:RH:forall x0 : R, f (g (f x0)) = f x0a, b:Rfa_eq_fb:f a = f ba = bf, g:R -> Rf_incr:forall x0 y : R, x0 < y -> f x0 < f yHyp:forall x0 : R, comp f g x0 = id x0x:RH:forall x0 : R, f (g (f x0)) = f x0f_inj:forall x0 y : R, f x0 = f y -> x0 = ycomp g f x = id xf, g:R -> Rf_incr:forall x0 y : R, x0 < y -> f x0 < f yHyp:forall x0 : R, comp f g x0 = id x0x:RH:forall x0 : R, f (g (f x0)) = f x0a, b:Rfa_eq_fb:f a = f b{a < b} + {a = b} -> a = bf, g:R -> Rf_incr:forall x0 y : R, x0 < y -> f x0 < f yHyp:forall x0 : R, comp f g x0 = id x0x:RH:forall x0 : R, f (g (f x0)) = f x0a, b:Rfa_eq_fb:f a = f ba > b -> a = bf, g:R -> Rf_incr:forall x0 y : R, x0 < y -> f x0 < f yHyp:forall x0 : R, comp f g x0 = id x0x:RH:forall x0 : R, f (g (f x0)) = f x0f_inj:forall x0 y : R, f x0 = f y -> x0 = ycomp g f x = id xf, g:R -> Rf_incr:forall x0 y : R, x0 < y -> f x0 < f yHyp:forall x0 : R, comp f g x0 = id x0x:RH:forall x0 : R, f (g (f x0)) = f x0a, b:Rfa_eq_fb:f a = f ba < b -> a = bf, g:R -> Rf_incr:forall x0 y : R, x0 < y -> f x0 < f yHyp:forall x0 : R, comp f g x0 = id x0x:RH:forall x0 : R, f (g (f x0)) = f x0a, b:Rfa_eq_fb:f a = f ba = b -> a = bf, g:R -> Rf_incr:forall x0 y : R, x0 < y -> f x0 < f yHyp:forall x0 : R, comp f g x0 = id x0x:RH:forall x0 : R, f (g (f x0)) = f x0a, b:Rfa_eq_fb:f a = f ba > b -> a = bf, g:R -> Rf_incr:forall x0 y : R, x0 < y -> f x0 < f yHyp:forall x0 : R, comp f g x0 = id x0x:RH:forall x0 : R, f (g (f x0)) = f x0f_inj:forall x0 y : R, f x0 = f y -> x0 = ycomp g f x = id xf, g:R -> Rf_incr:forall x0 y : R, x0 < y -> f x0 < f yHyp:forall x0 : R, comp f g x0 = id x0x:RH:forall x0 : R, f (g (f x0)) = f x0a, b:Rfa_eq_fb:f a = f bHf:a < ba = bf, g:R -> Rf_incr:forall x0 y : R, x0 < y -> f x0 < f yHyp:forall x0 : R, comp f g x0 = id x0x:RH:forall x0 : R, f (g (f x0)) = f x0a, b:Rfa_eq_fb:f a = f ba = b -> a = bf, g:R -> Rf_incr:forall x0 y : R, x0 < y -> f x0 < f yHyp:forall x0 : R, comp f g x0 = id x0x:RH:forall x0 : R, f (g (f x0)) = f x0a, b:Rfa_eq_fb:f a = f ba > b -> a = bf, g:R -> Rf_incr:forall x0 y : R, x0 < y -> f x0 < f yHyp:forall x0 : R, comp f g x0 = id x0x:RH:forall x0 : R, f (g (f x0)) = f x0f_inj:forall x0 y : R, f x0 = f y -> x0 = ycomp g f x = id xf, g:R -> Rf_incr:forall x0 y : R, x0 < y -> f x0 < f yHyp:forall x0 : R, comp f g x0 = id x0x:RH:forall x0 : R, f (g (f x0)) = f x0a, b:Rfa_eq_fb:f a = f bHf:a < bHfalse:f a < f ba = bf, g:R -> Rf_incr:forall x0 y : R, x0 < y -> f x0 < f yHyp:forall x0 : R, comp f g x0 = id x0x:RH:forall x0 : R, f (g (f x0)) = f x0a, b:Rfa_eq_fb:f a = f ba = b -> a = bf, g:R -> Rf_incr:forall x0 y : R, x0 < y -> f x0 < f yHyp:forall x0 : R, comp f g x0 = id x0x:RH:forall x0 : R, f (g (f x0)) = f x0a, b:Rfa_eq_fb:f a = f ba > b -> a = bf, g:R -> Rf_incr:forall x0 y : R, x0 < y -> f x0 < f yHyp:forall x0 : R, comp f g x0 = id x0x:RH:forall x0 : R, f (g (f x0)) = f x0f_inj:forall x0 y : R, f x0 = f y -> x0 = ycomp g f x = id xf, g:R -> Rf_incr:forall x0 y : R, x0 < y -> f x0 < f yHyp:forall x0 : R, comp f g x0 = id x0x:RH:forall x0 : R, f (g (f x0)) = f x0a, b:Rfa_eq_fb:f a = f bHf:a < bHfalse:f a < f bFalsef, g:R -> Rf_incr:forall x0 y : R, x0 < y -> f x0 < f yHyp:forall x0 : R, comp f g x0 = id x0x:RH:forall x0 : R, f (g (f x0)) = f x0a, b:Rfa_eq_fb:f a = f ba = b -> a = bf, g:R -> Rf_incr:forall x0 y : R, x0 < y -> f x0 < f yHyp:forall x0 : R, comp f g x0 = id x0x:RH:forall x0 : R, f (g (f x0)) = f x0a, b:Rfa_eq_fb:f a = f ba > b -> a = bf, g:R -> Rf_incr:forall x0 y : R, x0 < y -> f x0 < f yHyp:forall x0 : R, comp f g x0 = id x0x:RH:forall x0 : R, f (g (f x0)) = f x0f_inj:forall x0 y : R, f x0 = f y -> x0 = ycomp g f x = id xf, g:R -> Rf_incr:forall x0 y : R, x0 < y -> f x0 < f yHyp:forall x0 : R, comp f g x0 = id x0x:RH:forall x0 : R, f (g (f x0)) = f x0a, b:Rfa_eq_fb:f a = f ba = b -> a = bf, g:R -> Rf_incr:forall x0 y : R, x0 < y -> f x0 < f yHyp:forall x0 : R, comp f g x0 = id x0x:RH:forall x0 : R, f (g (f x0)) = f x0a, b:Rfa_eq_fb:f a = f ba > b -> a = bf, g:R -> Rf_incr:forall x0 y : R, x0 < y -> f x0 < f yHyp:forall x0 : R, comp f g x0 = id x0x:RH:forall x0 : R, f (g (f x0)) = f x0f_inj:forall x0 y : R, f x0 = f y -> x0 = ycomp g f x = id xf, g:R -> Rf_incr:forall x0 y : R, x0 < y -> f x0 < f yHyp:forall x0 : R, comp f g x0 = id x0x:RH:forall x0 : R, f (g (f x0)) = f x0a, b:Rfa_eq_fb:f a = f ba > b -> a = bf, g:R -> Rf_incr:forall x0 y : R, x0 < y -> f x0 < f yHyp:forall x0 : R, comp f g x0 = id x0x:RH:forall x0 : R, f (g (f x0)) = f x0f_inj:forall x0 y : R, f x0 = f y -> x0 = ycomp g f x = id xf, g:R -> Rf_incr:forall x0 y : R, x0 < y -> f x0 < f yHyp:forall x0 : R, comp f g x0 = id x0x:RH:forall x0 : R, f (g (f x0)) = f x0a, b:Rfa_eq_fb:f a = f bHf:a > ba = bf, g:R -> Rf_incr:forall x0 y : R, x0 < y -> f x0 < f yHyp:forall x0 : R, comp f g x0 = id x0x:RH:forall x0 : R, f (g (f x0)) = f x0f_inj:forall x0 y : R, f x0 = f y -> x0 = ycomp g f x = id xf, g:R -> Rf_incr:forall x0 y : R, x0 < y -> f x0 < f yHyp:forall x0 : R, comp f g x0 = id x0x:RH:forall x0 : R, f (g (f x0)) = f x0a, b:Rfa_eq_fb:f a = f bHf:a > bHfalse:f b < f aa = bf, g:R -> Rf_incr:forall x0 y : R, x0 < y -> f x0 < f yHyp:forall x0 : R, comp f g x0 = id x0x:RH:forall x0 : R, f (g (f x0)) = f x0f_inj:forall x0 y : R, f x0 = f y -> x0 = ycomp g f x = id xf, g:R -> Rf_incr:forall x0 y : R, x0 < y -> f x0 < f yHyp:forall x0 : R, comp f g x0 = id x0x:RH:forall x0 : R, f (g (f x0)) = f x0a, b:Rfa_eq_fb:f a = f bHf:a > bHfalse:f b < f aFalsef, g:R -> Rf_incr:forall x0 y : R, x0 < y -> f x0 < f yHyp:forall x0 : R, comp f g x0 = id x0x:RH:forall x0 : R, f (g (f x0)) = f x0f_inj:forall x0 y : R, f x0 = f y -> x0 = ycomp g f x = id xf, g:R -> Rf_incr:forall x0 y : R, x0 < y -> f x0 < f yHyp:forall x0 : R, comp f g x0 = id x0x:RH:forall x0 : R, f (g (f x0)) = f x0f_inj:forall x0 y : R, f x0 = f y -> x0 = ycomp g f x = id xf, g:R -> Rf_incr:forall x0 y : R, x0 < y -> f x0 < f yHyp:forall x0 : R, comp f g x0 = id x0x:RH:forall x0 : R, f (g (f x0)) = f x0f_inj:forall x0 y : R, f x0 = f y -> x0 = yf (comp g f x) = f (id x)f, g:R -> Rf_incr:forall x0 y : R, x0 < y -> f x0 < f yHyp:forall x0 : R, comp f g x0 = id x0x:RH:forall x0 : R, f (g (f x0)) = f x0f_inj:forall x0 y : R, f x0 = f y -> x0 = yf (g (f x)) = f (id x)f, g:R -> Rf_incr:forall x0 y : R, x0 < y -> f x0 < f yHyp:forall x0 : R, f (g x0) = id x0x:RH:forall x0 : R, f (g (f x0)) = f x0f_inj:forall x0 y : R, f x0 = f y -> x0 = yf (g (f x)) = f (id x)f, g:R -> Rf_incr:forall x0 y : R, x0 < y -> f x0 < f yHyp:forall x0 : R, f (g x0) = id x0x:RH:forall x0 : R, f (g (f x0)) = f x0f_inj:forall x0 y : R, f x0 = f y -> x0 = yid (f x) = f (id x)reflexivity. Qed. (* end hide *)f, g:R -> Rf_incr:forall x0 y : R, x0 < y -> f x0 < f yHyp:forall x0 : R, f (g x0) = id x0x:RH:forall x0 : R, f (g (f x0)) = f x0f_inj:forall x0 y : R, f x0 = f y -> x0 = yf x = f xforall (f g : R -> R) (lb ub : R), (forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f y) -> (forall y : R, f lb <= y -> y <= f ub -> comp f g y = id y) -> (forall x : R, f lb <= x -> x <= f ub -> lb <= g x <= ub) -> forall x : R, lb <= x <= ub -> comp g f x = id xforall (f g : R -> R) (lb ub : R), (forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f y) -> (forall y : R, f lb <= y -> y <= f ub -> comp f g y = id y) -> (forall x : R, f lb <= x -> x <= f ub -> lb <= g x <= ub) -> forall x : R, lb <= x <= ub -> comp g f x = id xf, g:R -> Rlb, ub:Rf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yHyp:forall y : R, f lb <= y -> y <= f ub -> comp f g y = id yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx:Rx_encad:lb <= x <= ubcomp g f x = id xf, g:R -> Rlb, ub:Rf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yHyp:forall y : R, f lb <= y -> y <= f ub -> comp f g y = id yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx:Rx_encad:lb <= x <= ubforall x0 y : R, lb <= x0 <= ub -> lb <= y <= ub -> f x0 = f y -> x0 = yf, g:R -> Rlb, ub:Rf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yHyp:forall y : R, f lb <= y -> y <= f ub -> comp f g y = id yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx:Rx_encad:lb <= x <= ubf_inj:forall x0 y : R, lb <= x0 <= ub -> lb <= y <= ub -> f x0 = f y -> x0 = ycomp g f x = id xf, g:R -> Rlb, ub:Rf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yHyp:forall y : R, f lb <= y -> y <= f ub -> comp f g y = id yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx:Rx_encad:lb <= x <= uba, b:Ra_encad:lb <= a <= ubb_encad:lb <= b <= ubfa_eq_fb:f a = f ba = bf, g:R -> Rlb, ub:Rf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yHyp:forall y : R, f lb <= y -> y <= f ub -> comp f g y = id yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx:Rx_encad:lb <= x <= ubf_inj:forall x0 y : R, lb <= x0 <= ub -> lb <= y <= ub -> f x0 = f y -> x0 = ycomp g f x = id xf, g:R -> Rlb, ub:Rf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yHyp:forall y : R, f lb <= y -> y <= f ub -> comp f g y = id yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx:Rx_encad:lb <= x <= uba, b:Ra_encad:lb <= a <= ubb_encad:lb <= b <= ubfa_eq_fb:f a = f b{a < b} + {a = b} -> a = bf, g:R -> Rlb, ub:Rf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yHyp:forall y : R, f lb <= y -> y <= f ub -> comp f g y = id yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx:Rx_encad:lb <= x <= uba, b:Ra_encad:lb <= a <= ubb_encad:lb <= b <= ubfa_eq_fb:f a = f ba > b -> a = bf, g:R -> Rlb, ub:Rf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yHyp:forall y : R, f lb <= y -> y <= f ub -> comp f g y = id yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx:Rx_encad:lb <= x <= ubf_inj:forall x0 y : R, lb <= x0 <= ub -> lb <= y <= ub -> f x0 = f y -> x0 = ycomp g f x = id xf, g:R -> Rlb, ub:Rf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yHyp:forall y : R, f lb <= y -> y <= f ub -> comp f g y = id yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx:Rx_encad:lb <= x <= uba, b:Ra_encad:lb <= a <= ubb_encad:lb <= b <= ubfa_eq_fb:f a = f ba < b -> a = bf, g:R -> Rlb, ub:Rf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yHyp:forall y : R, f lb <= y -> y <= f ub -> comp f g y = id yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx:Rx_encad:lb <= x <= uba, b:Ra_encad:lb <= a <= ubb_encad:lb <= b <= ubfa_eq_fb:f a = f ba = b -> a = bf, g:R -> Rlb, ub:Rf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yHyp:forall y : R, f lb <= y -> y <= f ub -> comp f g y = id yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx:Rx_encad:lb <= x <= uba, b:Ra_encad:lb <= a <= ubb_encad:lb <= b <= ubfa_eq_fb:f a = f ba > b -> a = bf, g:R -> Rlb, ub:Rf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yHyp:forall y : R, f lb <= y -> y <= f ub -> comp f g y = id yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx:Rx_encad:lb <= x <= ubf_inj:forall x0 y : R, lb <= x0 <= ub -> lb <= y <= ub -> f x0 = f y -> x0 = ycomp g f x = id xf, g:R -> Rlb, ub:Rf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yHyp:forall y : R, f lb <= y -> y <= f ub -> comp f g y = id yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx:Rx_encad:lb <= x <= uba, b:Ra_encad:lb <= a <= ubb_encad:lb <= b <= ubfa_eq_fb:f a = f bHf:a < ba = bf, g:R -> Rlb, ub:Rf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yHyp:forall y : R, f lb <= y -> y <= f ub -> comp f g y = id yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx:Rx_encad:lb <= x <= uba, b:Ra_encad:lb <= a <= ubb_encad:lb <= b <= ubfa_eq_fb:f a = f ba = b -> a = bf, g:R -> Rlb, ub:Rf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yHyp:forall y : R, f lb <= y -> y <= f ub -> comp f g y = id yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx:Rx_encad:lb <= x <= uba, b:Ra_encad:lb <= a <= ubb_encad:lb <= b <= ubfa_eq_fb:f a = f ba > b -> a = bf, g:R -> Rlb, ub:Rf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yHyp:forall y : R, f lb <= y -> y <= f ub -> comp f g y = id yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx:Rx_encad:lb <= x <= ubf_inj:forall x0 y : R, lb <= x0 <= ub -> lb <= y <= ub -> f x0 = f y -> x0 = ycomp g f x = id xf, g:R -> Rlb, ub:Rf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yHyp:forall y : R, f lb <= y -> y <= f ub -> comp f g y = id yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx:Rx_encad:lb <= x <= uba, b:Ra_encad:lb <= a <= ubb_encad:lb <= b <= ubfa_eq_fb:f a = f bHf:a < bHfalse:f a < f ba = bf, g:R -> Rlb, ub:Rf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yHyp:forall y : R, f lb <= y -> y <= f ub -> comp f g y = id yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx:Rx_encad:lb <= x <= uba, b:Ra_encad:lb <= a <= ubb_encad:lb <= b <= ubfa_eq_fb:f a = f ba = b -> a = bf, g:R -> Rlb, ub:Rf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yHyp:forall y : R, f lb <= y -> y <= f ub -> comp f g y = id yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx:Rx_encad:lb <= x <= uba, b:Ra_encad:lb <= a <= ubb_encad:lb <= b <= ubfa_eq_fb:f a = f ba > b -> a = bf, g:R -> Rlb, ub:Rf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yHyp:forall y : R, f lb <= y -> y <= f ub -> comp f g y = id yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx:Rx_encad:lb <= x <= ubf_inj:forall x0 y : R, lb <= x0 <= ub -> lb <= y <= ub -> f x0 = f y -> x0 = ycomp g f x = id xf, g:R -> Rlb, ub:Rf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yHyp:forall y : R, f lb <= y -> y <= f ub -> comp f g y = id yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx:Rx_encad:lb <= x <= uba, b:Ra_encad:lb <= a <= ubb_encad:lb <= b <= ubfa_eq_fb:f a = f bHf:a < bHfalse:f a < f bFalsef, g:R -> Rlb, ub:Rf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yHyp:forall y : R, f lb <= y -> y <= f ub -> comp f g y = id yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx:Rx_encad:lb <= x <= uba, b:Ra_encad:lb <= a <= ubb_encad:lb <= b <= ubfa_eq_fb:f a = f ba = b -> a = bf, g:R -> Rlb, ub:Rf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yHyp:forall y : R, f lb <= y -> y <= f ub -> comp f g y = id yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx:Rx_encad:lb <= x <= uba, b:Ra_encad:lb <= a <= ubb_encad:lb <= b <= ubfa_eq_fb:f a = f ba > b -> a = bf, g:R -> Rlb, ub:Rf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yHyp:forall y : R, f lb <= y -> y <= f ub -> comp f g y = id yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx:Rx_encad:lb <= x <= ubf_inj:forall x0 y : R, lb <= x0 <= ub -> lb <= y <= ub -> f x0 = f y -> x0 = ycomp g f x = id xf, g:R -> Rlb, ub:Rf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yHyp:forall y : R, f lb <= y -> y <= f ub -> comp f g y = id yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx:Rx_encad:lb <= x <= uba, b:Ra_encad:lb <= a <= ubb_encad:lb <= b <= ubfa_eq_fb:f a = f ba = b -> a = bf, g:R -> Rlb, ub:Rf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yHyp:forall y : R, f lb <= y -> y <= f ub -> comp f g y = id yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx:Rx_encad:lb <= x <= uba, b:Ra_encad:lb <= a <= ubb_encad:lb <= b <= ubfa_eq_fb:f a = f ba > b -> a = bf, g:R -> Rlb, ub:Rf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yHyp:forall y : R, f lb <= y -> y <= f ub -> comp f g y = id yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx:Rx_encad:lb <= x <= ubf_inj:forall x0 y : R, lb <= x0 <= ub -> lb <= y <= ub -> f x0 = f y -> x0 = ycomp g f x = id xf, g:R -> Rlb, ub:Rf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yHyp:forall y : R, f lb <= y -> y <= f ub -> comp f g y = id yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx:Rx_encad:lb <= x <= uba, b:Ra_encad:lb <= a <= ubb_encad:lb <= b <= ubfa_eq_fb:f a = f ba > b -> a = bf, g:R -> Rlb, ub:Rf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yHyp:forall y : R, f lb <= y -> y <= f ub -> comp f g y = id yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx:Rx_encad:lb <= x <= ubf_inj:forall x0 y : R, lb <= x0 <= ub -> lb <= y <= ub -> f x0 = f y -> x0 = ycomp g f x = id xf, g:R -> Rlb, ub:Rf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yHyp:forall y : R, f lb <= y -> y <= f ub -> comp f g y = id yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx:Rx_encad:lb <= x <= uba, b:Ra_encad:lb <= a <= ubb_encad:lb <= b <= ubfa_eq_fb:f a = f bHf:a > ba = bf, g:R -> Rlb, ub:Rf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yHyp:forall y : R, f lb <= y -> y <= f ub -> comp f g y = id yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx:Rx_encad:lb <= x <= ubf_inj:forall x0 y : R, lb <= x0 <= ub -> lb <= y <= ub -> f x0 = f y -> x0 = ycomp g f x = id xf, g:R -> Rlb, ub:Rf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yHyp:forall y : R, f lb <= y -> y <= f ub -> comp f g y = id yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx:Rx_encad:lb <= x <= uba, b:Ra_encad:lb <= a <= ubb_encad:lb <= b <= ubfa_eq_fb:f a = f bHf:a > bHfalse:f b < f aa = bf, g:R -> Rlb, ub:Rf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yHyp:forall y : R, f lb <= y -> y <= f ub -> comp f g y = id yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx:Rx_encad:lb <= x <= ubf_inj:forall x0 y : R, lb <= x0 <= ub -> lb <= y <= ub -> f x0 = f y -> x0 = ycomp g f x = id xf, g:R -> Rlb, ub:Rf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yHyp:forall y : R, f lb <= y -> y <= f ub -> comp f g y = id yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx:Rx_encad:lb <= x <= uba, b:Ra_encad:lb <= a <= ubb_encad:lb <= b <= ubfa_eq_fb:f a = f bHf:a > bHfalse:f b < f aFalsef, g:R -> Rlb, ub:Rf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yHyp:forall y : R, f lb <= y -> y <= f ub -> comp f g y = id yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx:Rx_encad:lb <= x <= ubf_inj:forall x0 y : R, lb <= x0 <= ub -> lb <= y <= ub -> f x0 = f y -> x0 = ycomp g f x = id xf, g:R -> Rlb, ub:Rf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yHyp:forall y : R, f lb <= y -> y <= f ub -> comp f g y = id yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx:Rx_encad:lb <= x <= ubf_inj:forall x0 y : R, lb <= x0 <= ub -> lb <= y <= ub -> f x0 = f y -> x0 = ycomp g f x = id xf, g:R -> Rlb, ub:Rf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yHyp:forall y : R, f lb <= y -> y <= f ub -> comp f g y = id yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx:Rx_encad:lb <= x <= ubf_inj:forall x0 y : R, lb <= x0 <= ub -> lb <= y <= ub -> f x0 = f y -> x0 = yforall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yf, g:R -> Rlb, ub:Rf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yHyp:forall y : R, f lb <= y -> y <= f ub -> comp f g y = id yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx:Rx_encad:lb <= x <= ubf_inj:forall x0 y : R, lb <= x0 <= ub -> lb <= y <= ub -> f x0 = f y -> x0 = yf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f ycomp g f x = id xf, g:R -> Rlb, ub:Rf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yHyp:forall y : R, f lb <= y -> y <= f ub -> comp f g y = id yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx:Rx_encad:lb <= x <= ubf_inj:forall x0 y : R, lb <= x0 <= ub -> lb <= y <= ub -> f x0 = f y -> x0 = ym, n:Rcond1:lb <= mcond2:m <= ncond3:n <= ubf m <= f nf, g:R -> Rlb, ub:Rf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yHyp:forall y : R, f lb <= y -> y <= f ub -> comp f g y = id yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx:Rx_encad:lb <= x <= ubf_inj:forall x0 y : R, lb <= x0 <= ub -> lb <= y <= ub -> f x0 = f y -> x0 = yf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f ycomp g f x = id xf, g:R -> Rlb, ub:Rf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yHyp:forall y : R, f lb <= y -> y <= f ub -> comp f g y = id yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx:Rx_encad:lb <= x <= ubf_inj:forall x0 y : R, lb <= x0 <= ub -> lb <= y <= ub -> f x0 = f y -> x0 = ym, n:Rcond1:lb <= mcond2:m <= ncond3:n <= ubm < n -> f m <= f nf, g:R -> Rlb, ub:Rf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yHyp:forall y : R, f lb <= y -> y <= f ub -> comp f g y = id yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx:Rx_encad:lb <= x <= ubf_inj:forall x0 y : R, lb <= x0 <= ub -> lb <= y <= ub -> f x0 = f y -> x0 = ym, n:Rcond1:lb <= mcond2:m <= ncond3:n <= ubm = n -> f m <= f nf, g:R -> Rlb, ub:Rf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yHyp:forall y : R, f lb <= y -> y <= f ub -> comp f g y = id yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx:Rx_encad:lb <= x <= ubf_inj:forall x0 y : R, lb <= x0 <= ub -> lb <= y <= ub -> f x0 = f y -> x0 = yf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f ycomp g f x = id xf, g:R -> Rlb, ub:Rf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yHyp:forall y : R, f lb <= y -> y <= f ub -> comp f g y = id yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx:Rx_encad:lb <= x <= ubf_inj:forall x0 y : R, lb <= x0 <= ub -> lb <= y <= ub -> f x0 = f y -> x0 = ym, n:Rcond1:lb <= mcond2:m <= ncond3:n <= ubcond:m < nf m <= f nf, g:R -> Rlb, ub:Rf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yHyp:forall y : R, f lb <= y -> y <= f ub -> comp f g y = id yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx:Rx_encad:lb <= x <= ubf_inj:forall x0 y : R, lb <= x0 <= ub -> lb <= y <= ub -> f x0 = f y -> x0 = ym, n:Rcond1:lb <= mcond2:m <= ncond3:n <= ubm = n -> f m <= f nf, g:R -> Rlb, ub:Rf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yHyp:forall y : R, f lb <= y -> y <= f ub -> comp f g y = id yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx:Rx_encad:lb <= x <= ubf_inj:forall x0 y : R, lb <= x0 <= ub -> lb <= y <= ub -> f x0 = f y -> x0 = yf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f ycomp g f x = id xf, g:R -> Rlb, ub:Rf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yHyp:forall y : R, f lb <= y -> y <= f ub -> comp f g y = id yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx:Rx_encad:lb <= x <= ubf_inj:forall x0 y : R, lb <= x0 <= ub -> lb <= y <= ub -> f x0 = f y -> x0 = ym, n:Rcond1:lb <= mcond2:m <= ncond3:n <= ubm = n -> f m <= f nf, g:R -> Rlb, ub:Rf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yHyp:forall y : R, f lb <= y -> y <= f ub -> comp f g y = id yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx:Rx_encad:lb <= x <= ubf_inj:forall x0 y : R, lb <= x0 <= ub -> lb <= y <= ub -> f x0 = f y -> x0 = yf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f ycomp g f x = id xf, g:R -> Rlb, ub:Rf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yHyp:forall y : R, f lb <= y -> y <= f ub -> comp f g y = id yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx:Rx_encad:lb <= x <= ubf_inj:forall x0 y : R, lb <= x0 <= ub -> lb <= y <= ub -> f x0 = f y -> x0 = yf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f ycomp g f x = id xf, g:R -> Rlb, ub:Rf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yHyp:forall y : R, f lb <= y -> y <= f ub -> comp f g y = id yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx:Rx_encad:lb <= x <= ubf_inj:forall x0 y : R, lb <= x0 <= ub -> lb <= y <= ub -> f x0 = f y -> x0 = yf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yforall x0 : R, lb <= x0 <= ub -> f (g (f x0)) = f x0f, g:R -> Rlb, ub:Rf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yHyp:forall y : R, f lb <= y -> y <= f ub -> comp f g y = id yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx:Rx_encad:lb <= x <= ubf_inj:forall x0 y : R, lb <= x0 <= ub -> lb <= y <= ub -> f x0 = f y -> x0 = yf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yHyp2:forall x0 : R, lb <= x0 <= ub -> f (g (f x0)) = f x0comp g f x = id xf, g:R -> Rlb, ub:Rf_incr_interv:forall x1 y : R, lb <= x1 -> x1 < y -> y <= ub -> f x1 < f yHyp:forall y : R, f lb <= y -> y <= f ub -> comp f g y = id yg_wf:forall x1 : R, f lb <= x1 -> x1 <= f ub -> lb <= g x1 <= ubx:Rx_encad:lb <= x <= ubf_inj:forall x1 y : R, lb <= x1 <= ub -> lb <= y <= ub -> f x1 = f y -> x1 = yf_incr_interv2:forall x1 y : R, lb <= x1 -> x1 <= y -> y <= ub -> f x1 <= f yx0:RH:lb <= x0 <= ubf lb <= f x0f, g:R -> Rlb, ub:Rf_incr_interv:forall x1 y : R, lb <= x1 -> x1 < y -> y <= ub -> f x1 < f yHyp:forall y : R, f lb <= y -> y <= f ub -> comp f g y = id yg_wf:forall x1 : R, f lb <= x1 -> x1 <= f ub -> lb <= g x1 <= ubx:Rx_encad:lb <= x <= ubf_inj:forall x1 y : R, lb <= x1 <= ub -> lb <= y <= ub -> f x1 = f y -> x1 = yf_incr_interv2:forall x1 y : R, lb <= x1 -> x1 <= y -> y <= ub -> f x1 <= f yx0:RH:lb <= x0 <= ubf x0 <= f ubf, g:R -> Rlb, ub:Rf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yHyp:forall y : R, f lb <= y -> y <= f ub -> comp f g y = id yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx:Rx_encad:lb <= x <= ubf_inj:forall x0 y : R, lb <= x0 <= ub -> lb <= y <= ub -> f x0 = f y -> x0 = yf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yHyp2:forall x0 : R, lb <= x0 <= ub -> f (g (f x0)) = f x0comp g f x = id xf, g:R -> Rlb, ub:Rf_incr_interv:forall x1 y : R, lb <= x1 -> x1 < y -> y <= ub -> f x1 < f yHyp:forall y : R, f lb <= y -> y <= f ub -> comp f g y = id yg_wf:forall x1 : R, f lb <= x1 -> x1 <= f ub -> lb <= g x1 <= ubx:Rx_encad:lb <= x <= ubf_inj:forall x1 y : R, lb <= x1 <= ub -> lb <= y <= ub -> f x1 = f y -> x1 = yf_incr_interv2:forall x1 y : R, lb <= x1 -> x1 <= y -> y <= ub -> f x1 <= f yx0:RH:lb <= x0 <= ubf x0 <= f ubf, g:R -> Rlb, ub:Rf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yHyp:forall y : R, f lb <= y -> y <= f ub -> comp f g y = id yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx:Rx_encad:lb <= x <= ubf_inj:forall x0 y : R, lb <= x0 <= ub -> lb <= y <= ub -> f x0 = f y -> x0 = yf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yHyp2:forall x0 : R, lb <= x0 <= ub -> f (g (f x0)) = f x0comp g f x = id xf, g:R -> Rlb, ub:Rf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yHyp:forall y : R, f lb <= y -> y <= f ub -> comp f g y = id yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx:Rx_encad:lb <= x <= ubf_inj:forall x0 y : R, lb <= x0 <= ub -> lb <= y <= ub -> f x0 = f y -> x0 = yf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yHyp2:forall x0 : R, lb <= x0 <= ub -> f (g (f x0)) = f x0comp g f x = id xf, g:R -> Rlb, ub:Rf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yHyp:forall y : R, f lb <= y -> y <= f ub -> f (g y) = id yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx:Rx_encad:lb <= x <= ubf_inj:forall x0 y : R, lb <= x0 <= ub -> lb <= y <= ub -> f x0 = f y -> x0 = yf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yHyp2:forall x0 : R, lb <= x0 <= ub -> f (g (f x0)) = f x0g (f x) = id xf, g:R -> Rlb, ub:Rf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yHyp:forall y : R, f lb <= y -> y <= f ub -> f (g y) = id yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx:Rx_encad:lb <= x <= ubf_inj:forall x0 y : R, lb <= x0 <= ub -> lb <= y <= ub -> f x0 = f y -> x0 = yf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yHyp2:forall x0 : R, lb <= x0 <= ub -> f (g (f x0)) = f x0lb <= g (f x) <= ubf, g:R -> Rlb, ub:Rf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yHyp:forall y : R, f lb <= y -> y <= f ub -> f (g y) = id yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx:Rx_encad:lb <= x <= ubf_inj:forall x0 y : R, lb <= x0 <= ub -> lb <= y <= ub -> f x0 = f y -> x0 = yf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yHyp2:forall x0 : R, lb <= x0 <= ub -> f (g (f x0)) = f x0lb <= id x <= ubf, g:R -> Rlb, ub:Rf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yHyp:forall y : R, f lb <= y -> y <= f ub -> f (g y) = id yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx:Rx_encad:lb <= x <= ubf_inj:forall x0 y : R, lb <= x0 <= ub -> lb <= y <= ub -> f x0 = f y -> x0 = yf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yHyp2:forall x0 : R, lb <= x0 <= ub -> f (g (f x0)) = f x0f (g (f x)) = f (id x)f, g:R -> Rlb, ub:Rf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yHyp:forall y : R, f lb <= y -> y <= f ub -> f (g y) = id yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx:Rx_encad:lb <= x <= ubf_inj:forall x0 y : R, lb <= x0 <= ub -> lb <= y <= ub -> f x0 = f y -> x0 = yf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yHyp2:forall x0 : R, lb <= x0 <= ub -> f (g (f x0)) = f x0lb <= id x <= ubf, g:R -> Rlb, ub:Rf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yHyp:forall y : R, f lb <= y -> y <= f ub -> f (g y) = id yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx:Rx_encad:lb <= x <= ubf_inj:forall x0 y : R, lb <= x0 <= ub -> lb <= y <= ub -> f x0 = f y -> x0 = yf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yHyp2:forall x0 : R, lb <= x0 <= ub -> f (g (f x0)) = f x0f (g (f x)) = f (id x)apply Hyp2 ; unfold id ; assumption. Qed.f, g:R -> Rlb, ub:Rf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yHyp:forall y : R, f lb <= y -> y <= f ub -> f (g y) = id yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubx:Rx_encad:lb <= x <= ubf_inj:forall x0 y : R, lb <= x0 <= ub -> lb <= y <= ub -> f x0 = f y -> x0 = yf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yHyp2:forall x0 : R, lb <= x0 <= ub -> f (g (f x0)) = f x0f (g (f x)) = f (id x)
Intermediate Value Theorem on an Interval (Proof mainly taken from Reals.Rsqrt_def) and its corollary
forall (x y : R) (P : R -> bool) (N : nat), x < y -> x <= Dichotomy_ub x y P N <= y /\ x <= Dichotomy_lb x y P N <= yforall (x y : R) (P : R -> bool) (N : nat), x < y -> x <= Dichotomy_ub x y P N <= y /\ x <= Dichotomy_lb x y P N <= yforall x y lb ub : R, lb <= x <= ub /\ lb <= y <= ub -> lb <= (x + y) / 2 <= ubSublemma:forall x y lb ub : R, lb <= x <= ub /\ lb <= y <= ub -> lb <= (x + y) / 2 <= ubforall (x y : R) (P : R -> bool) (N : nat), x < y -> x <= Dichotomy_ub x y P N <= y /\ x <= Dichotomy_lb x y P N <= yx, y, lb, ub:RHyp:lb <= x <= ub /\ lb <= y <= ublb <= (x + y) / 2 <= ubSublemma:forall x y lb ub : R, lb <= x <= ub /\ lb <= y <= ub -> lb <= (x + y) / 2 <= ubforall (x y : R) (P : R -> bool) (N : nat), x < y -> x <= Dichotomy_ub x y P N <= y /\ x <= Dichotomy_lb x y P N <= ySublemma:forall x y lb ub : R, lb <= x <= ub /\ lb <= y <= ub -> lb <= (x + y) / 2 <= ubforall (x y : R) (P : R -> bool) (N : nat), x < y -> x <= Dichotomy_ub x y P N <= y /\ x <= Dichotomy_lb x y P N <= ySublemma:forall x0 y0 lb ub : R, lb <= x0 <= ub /\ lb <= y0 <= ub -> lb <= (x0 + y0) / 2 <= ubx, y:RP:R -> boolN:natx_lt_y:x < yx <= Dichotomy_ub x y P N <= y /\ x <= Dichotomy_lb x y P N <= ySublemma:forall x0 y0 lb ub : R, lb <= x0 <= ub /\ lb <= y0 <= ub -> lb <= (x0 + y0) / 2 <= ubx, y:RP:R -> boolx_lt_y:x < yx <= Dichotomy_ub x y P 0 <= y /\ x <= Dichotomy_lb x y P 0 <= ySublemma:forall x0 y0 lb ub : R, lb <= x0 <= ub /\ lb <= y0 <= ub -> lb <= (x0 + y0) / 2 <= ubx, y:RP:R -> boolN:natx_lt_y:x < yIHN:x <= Dichotomy_ub x y P N <= y /\ x <= Dichotomy_lb x y P N <= yx <= Dichotomy_ub x y P (S N) <= y /\ x <= Dichotomy_lb x y P (S N) <= ySublemma:forall x0 y0 lb ub : R, lb <= x0 <= ub /\ lb <= y0 <= ub -> lb <= (x0 + y0) / 2 <= ubx, y:RP:R -> boolN:natx_lt_y:x < yIHN:x <= Dichotomy_ub x y P N <= y /\ x <= Dichotomy_lb x y P N <= yx <= Dichotomy_ub x y P (S N) <= y /\ x <= Dichotomy_lb x y P (S N) <= ySublemma:forall x0 y0 lb ub : R, lb <= x0 <= ub /\ lb <= y0 <= ub -> lb <= (x0 + y0) / 2 <= ubx, y:RP:R -> boolN:natx_lt_y:x < yIHN:x <= Dichotomy_ub x y P N <= y /\ x <= Dichotomy_lb x y P N <= yx <= (if P ((Dichotomy_lb x y P N + Dichotomy_ub x y P N) / 2) then (Dichotomy_lb x y P N + Dichotomy_ub x y P N) / 2 else Dichotomy_ub x y P N) <= y /\ x <= (if P ((Dichotomy_lb x y P N + Dichotomy_ub x y P N) / 2) then Dichotomy_lb x y P N else (Dichotomy_lb x y P N + Dichotomy_ub x y P N) / 2) <= ySublemma:forall x0 y0 lb ub : R, lb <= x0 <= ub /\ lb <= y0 <= ub -> lb <= (x0 + y0) / 2 <= ubx, y:RP:R -> boolN:natx_lt_y:x < yIHN:x <= Dichotomy_ub x y P N <= y /\ x <= Dichotomy_lb x y P N <= yx <= (Dichotomy_lb x y P N + Dichotomy_ub x y P N) / 2 <= y /\ x <= Dichotomy_lb x y P N <= ySublemma:forall x0 y0 lb ub : R, lb <= x0 <= ub /\ lb <= y0 <= ub -> lb <= (x0 + y0) / 2 <= ubx, y:RP:R -> boolN:natx_lt_y:x < yIHN:x <= Dichotomy_ub x y P N <= y /\ x <= Dichotomy_lb x y P N <= yx <= Dichotomy_ub x y P N <= y /\ x <= (Dichotomy_lb x y P N + Dichotomy_ub x y P N) / 2 <= ySublemma:forall x0 y0 lb ub : R, lb <= x0 <= ub /\ lb <= y0 <= ub -> lb <= (x0 + y0) / 2 <= ubx, y:RP:R -> boolN:natx_lt_y:x < yIHN:x <= Dichotomy_ub x y P N <= y /\ x <= Dichotomy_lb x y P N <= yx <= (Dichotomy_lb x y P N + Dichotomy_ub x y P N) / 2 <= ySublemma:forall x0 y0 lb ub : R, lb <= x0 <= ub /\ lb <= y0 <= ub -> lb <= (x0 + y0) / 2 <= ubx, y:RP:R -> boolN:natx_lt_y:x < yIHN:x <= Dichotomy_ub x y P N <= y /\ x <= Dichotomy_lb x y P N <= yx <= Dichotomy_lb x y P N <= ySublemma:forall x0 y0 lb ub : R, lb <= x0 <= ub /\ lb <= y0 <= ub -> lb <= (x0 + y0) / 2 <= ubx, y:RP:R -> boolN:natx_lt_y:x < yIHN:x <= Dichotomy_ub x y P N <= y /\ x <= Dichotomy_lb x y P N <= yx <= Dichotomy_ub x y P N <= y /\ x <= (Dichotomy_lb x y P N + Dichotomy_ub x y P N) / 2 <= ySublemma:forall x0 y0 lb ub : R, lb <= x0 <= ub /\ lb <= y0 <= ub -> lb <= (x0 + y0) / 2 <= ubx, y:RP:R -> boolN:natx_lt_y:x < yIHN:x <= Dichotomy_ub x y P N <= y /\ x <= Dichotomy_lb x y P N <= yx <= Dichotomy_lb x y P N <= ySublemma:forall x0 y0 lb ub : R, lb <= x0 <= ub /\ lb <= y0 <= ub -> lb <= (x0 + y0) / 2 <= ubx, y:RP:R -> boolN:natx_lt_y:x < yIHN:x <= Dichotomy_ub x y P N <= y /\ x <= Dichotomy_lb x y P N <= yx <= Dichotomy_ub x y P N <= y /\ x <= (Dichotomy_lb x y P N + Dichotomy_ub x y P N) / 2 <= ySublemma:forall x0 y0 lb ub : R, lb <= x0 <= ub /\ lb <= y0 <= ub -> lb <= (x0 + y0) / 2 <= ubx, y:RP:R -> boolN:natx_lt_y:x < yIHN:x <= Dichotomy_ub x y P N <= y /\ x <= Dichotomy_lb x y P N <= yx <= Dichotomy_ub x y P N <= y /\ x <= (Dichotomy_lb x y P N + Dichotomy_ub x y P N) / 2 <= ySublemma:forall x0 y0 lb ub : R, lb <= x0 <= ub /\ lb <= y0 <= ub -> lb <= (x0 + y0) / 2 <= ubx, y:RP:R -> boolN:natx_lt_y:x < yIHN:x <= Dichotomy_ub x y P N <= y /\ x <= Dichotomy_lb x y P N <= yx <= Dichotomy_ub x y P N <= ySublemma:forall x0 y0 lb ub : R, lb <= x0 <= ub /\ lb <= y0 <= ub -> lb <= (x0 + y0) / 2 <= ubx, y:RP:R -> boolN:natx_lt_y:x < yIHN:x <= Dichotomy_ub x y P N <= y /\ x <= Dichotomy_lb x y P N <= yx <= (Dichotomy_lb x y P N + Dichotomy_ub x y P N) / 2 <= yapply Sublemma ; intuition. Qed.Sublemma:forall x0 y0 lb ub : R, lb <= x0 <= ub /\ lb <= y0 <= ub -> lb <= (x0 + y0) / 2 <= ubx, y:RP:R -> boolN:natx_lt_y:x < yIHN:x <= Dichotomy_ub x y P N <= y /\ x <= Dichotomy_lb x y P N <= yx <= (Dichotomy_lb x y P N + Dichotomy_ub x y P N) / 2 <= yforall (x y x0 : R) (D : R -> bool), x < y -> Un_cv (dicho_up x y D) x0 -> x <= x0 <= yforall (x y x0 : R) (D : R -> bool), x < y -> Un_cv (dicho_up x y D) x0 -> x <= x0 <= yx, y, x0:RD:R -> boolx_lt_y:x < ybnd:Un_cv (dicho_up x y D) x0x <= x0 <= yx, y, x0:RD:R -> boolx_lt_y:x < ybnd:Un_cv (dicho_up x y D) x0forall n : nat, x <= dicho_up x y D n <= yx, y, x0:RD:R -> boolx_lt_y:x < ybnd:Un_cv (dicho_up x y D) x0Main:forall n : nat, x <= dicho_up x y D n <= yx <= x0 <= yx, y, x0:RD:R -> boolx_lt_y:x < ybnd:Un_cv (dicho_up x y D) x0n:natx <= dicho_up x y D n <= yx, y, x0:RD:R -> boolx_lt_y:x < ybnd:Un_cv (dicho_up x y D) x0Main:forall n : nat, x <= dicho_up x y D n <= yx <= x0 <= yx, y, x0:RD:R -> boolx_lt_y:x < ybnd:Un_cv (dicho_up x y D) x0n:natx <= Dichotomy_ub x y D n <= yx, y, x0:RD:R -> boolx_lt_y:x < ybnd:Un_cv (dicho_up x y D) x0Main:forall n : nat, x <= dicho_up x y D n <= yx <= x0 <= yx, y, x0:RD:R -> boolx_lt_y:x < ybnd:Un_cv (dicho_up x y D) x0Main:forall n : nat, x <= dicho_up x y D n <= yx <= x0 <= yx, y, x0:RD:R -> boolx_lt_y:x < ybnd:Un_cv (dicho_up x y D) x0Main:forall n : nat, x <= dicho_up x y D n <= yx <= x0x, y, x0:RD:R -> boolx_lt_y:x < ybnd:Un_cv (dicho_up x y D) x0Main:forall n : nat, x <= dicho_up x y D n <= yx0 <= yx, y, x0:RD:R -> boolx_lt_y:x < ybnd:Un_cv (dicho_up x y D) x0Main:forall n : nat, x <= dicho_up x y D n <= yforall n : nat, x <= dicho_up x y D nx, y, x0:RD:R -> boolx_lt_y:x < ybnd:Un_cv (dicho_up x y D) x0Main:forall n : nat, x <= dicho_up x y D n <= yUn_cv (fun _ : nat => x) xx, y, x0:RD:R -> boolx_lt_y:x < ybnd:Un_cv (dicho_up x y D) x0Main:forall n : nat, x <= dicho_up x y D n <= yUn_cv (dicho_up x y D) x0x, y, x0:RD:R -> boolx_lt_y:x < ybnd:Un_cv (dicho_up x y D) x0Main:forall n : nat, x <= dicho_up x y D n <= yx0 <= yx, y, x0:RD:R -> boolx_lt_y:x < ybnd:Un_cv (dicho_up x y D) x0Main:forall n : nat, x <= dicho_up x y D n <= yUn_cv (fun _ : nat => x) xx, y, x0:RD:R -> boolx_lt_y:x < ybnd:Un_cv (dicho_up x y D) x0Main:forall n : nat, x <= dicho_up x y D n <= yUn_cv (dicho_up x y D) x0x, y, x0:RD:R -> boolx_lt_y:x < ybnd:Un_cv (dicho_up x y D) x0Main:forall n : nat, x <= dicho_up x y D n <= yx0 <= yx, y, x0:RD:R -> boolx_lt_y:x < ybnd:Un_cv (dicho_up x y D) x0Main:forall n : nat, x <= dicho_up x y D n <= yUn_cv (dicho_up x y D) x0x, y, x0:RD:R -> boolx_lt_y:x < ybnd:Un_cv (dicho_up x y D) x0Main:forall n : nat, x <= dicho_up x y D n <= yx0 <= yx, y, x0:RD:R -> boolx_lt_y:x < ybnd:Un_cv (dicho_up x y D) x0Main:forall n : nat, x <= dicho_up x y D n <= yx0 <= yx, y, x0:RD:R -> boolx_lt_y:x < ybnd:Un_cv (dicho_up x y D) x0Main:forall n : nat, x <= dicho_up x y D n <= yforall n : nat, dicho_up x y D n <= yx, y, x0:RD:R -> boolx_lt_y:x < ybnd:Un_cv (dicho_up x y D) x0Main:forall n : nat, x <= dicho_up x y D n <= yUn_cv (dicho_up x y D) x0x, y, x0:RD:R -> boolx_lt_y:x < ybnd:Un_cv (dicho_up x y D) x0Main:forall n : nat, x <= dicho_up x y D n <= yUn_cv (fun _ : nat => y) yx, y, x0:RD:R -> boolx_lt_y:x < ybnd:Un_cv (dicho_up x y D) x0Main:forall n : nat, x <= dicho_up x y D n <= yUn_cv (dicho_up x y D) x0x, y, x0:RD:R -> boolx_lt_y:x < ybnd:Un_cv (dicho_up x y D) x0Main:forall n : nat, x <= dicho_up x y D n <= yUn_cv (fun _ : nat => y) yunfold Un_cv ; intros ; exists 0%nat ; intros ; unfold R_dist ; replace (y -y) with 0 by field ; rewrite Rabs_R0 ; assumption. Qed.x, y, x0:RD:R -> boolx_lt_y:x < ybnd:Un_cv (dicho_up x y D) x0Main:forall n : nat, x <= dicho_up x y D n <= yUn_cv (fun _ : nat => y) yforall (f : R -> R) (x y : R), (forall a : R, x <= a <= y -> continuity_pt f a) -> x < y -> f x < 0 -> 0 < f y -> {z : R | x <= z <= y /\ f z = 0}forall (f : R -> R) (x y : R), (forall a : R, x <= a <= y -> continuity_pt f a) -> x < y -> f x < 0 -> 0 < f y -> {z : R | x <= z <= y /\ f z = 0}f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f y{z : R | x <= z <= y /\ f z = 0}f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yx <= y -> {z : R | x <= z <= y /\ f z = 0}f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yx <= yf:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= y{z : R | x <= z <= y /\ f z = 0}f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yx <= yf:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= y{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l} -> {z : R | x <= z <= y /\ f z = 0}f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yx <= yf:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= y{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l} -> {l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l} -> {z : R | x <= z <= y /\ f z = 0}f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yx <= yf:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}{z : R | x <= z <= y /\ f z = 0}f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yx <= yf:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0{z : R | x <= z <= y /\ f z = 0}f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yx <= yf:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x1{z : R | x <= z <= y /\ f z = 0}f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yx <= yf:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x1H4:x1 = x0{z : R | x <= z <= y /\ f z = 0}f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yx <= yf:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0{z : R | x <= z <= y /\ f z = 0}f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yx <= yf:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0x <= x0 <= y /\ f x0 = 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yx <= yf:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0x <= x0 <= yf:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0f x0 = 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yx <= yf:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0x <= x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0x0 <= yf:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0f x0 = 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yx <= yf:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0x <= dicho_lb x y (fun z : R => cond_positivity (f z)) 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0dicho_lb x y (fun z : R => cond_positivity (f z)) 0 <= x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0x0 <= yf:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0f x0 = 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yx <= yf:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0x <= xf:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0dicho_lb x y (fun z : R => cond_positivity (f z)) 0 <= x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0x0 <= yf:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0f x0 = 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yx <= yf:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0dicho_lb x y (fun z : R => cond_positivity (f z)) 0 <= x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0x0 <= yf:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0f x0 = 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yx <= yf:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Un_growing (dicho_lb x y (fun z : R => cond_positivity (f z)))f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0x0 <= yf:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0f x0 = 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yx <= yf:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0x0 <= yf:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0f x0 = 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yx <= yf:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0x0 <= yf:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0f x0 = 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yx <= yf:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0x0 <= dicho_up x y (fun z : R => cond_positivity (f z)) 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0dicho_up x y (fun z : R => cond_positivity (f z)) 0 <= yf:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0f x0 = 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yx <= yf:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Un_decreasing (dicho_up x y (fun z : R => cond_positivity (f z)))f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0dicho_up x y (fun z : R => cond_positivity (f z)) 0 <= yf:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0f x0 = 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yx <= yf:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0dicho_up x y (fun z : R => cond_positivity (f z)) 0 <= yf:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0f x0 = 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yx <= yf:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0dicho_up x y (fun z : R => cond_positivity (f z)) 0 <= yf:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0f x0 = 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yx <= yf:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0f x0 = 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yx <= yf:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0f x0 = 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> Rf x0 = 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> Rf x0 = 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R((forall n : nat, f (Vn n) <= 0) -> f x0 <= 0) -> f x0 = 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R((forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0) -> ((forall n : nat, f (Vn n) <= 0) -> f x0 <= 0) -> f x0 = 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0f x0 = 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0(forall n : nat, f (Vn n) <= 0) -> f x0 = 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0forall n : nat, f (Vn n) <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0(forall n : nat, 0 <= f (Wn n)) -> (forall n : nat, f (Vn n) <= 0) -> f x0 = 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0forall n : nat, 0 <= f (Wn n)f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0forall n : nat, f (Vn n) <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0H7:forall n : nat, 0 <= f (Wn n)H8:forall n : nat, f (Vn n) <= 0f x0 = 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0forall n : nat, 0 <= f (Wn n)f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0forall n : nat, f (Vn n) <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0H7:forall n : nat, 0 <= f (Wn n)H8:forall n : nat, f (Vn n) <= 0H9:f x0 <= 0f x0 = 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0forall n : nat, 0 <= f (Wn n)f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0forall n : nat, f (Vn n) <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0H7:forall n : nat, 0 <= f (Wn n)H8:forall n : nat, f (Vn n) <= 0H9:f x0 <= 0H10:0 <= f x0f x0 = 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0forall n : nat, 0 <= f (Wn n)f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0forall n : nat, f (Vn n) <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0forall n : nat, 0 <= f (Wn n)f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0forall n : nat, f (Vn n) <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n0 : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n0:nat -> RWn:=fun n0 : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n0:nat -> RH5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0n:nat0 <= f (Wn n)f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0forall n : nat, f (Vn n) <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n0 : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n0:nat -> RWn:=fun n0 : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n0:nat -> RH5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0n:nat0 <= f (dicho_up x y (fun z : R => cond_positivity (f z)) n)f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0forall n : nat, f (Vn n) <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n0 : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n0:nat -> RWn:=fun n0 : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n0:nat -> RH5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0n:nat(forall z : R, cond_positivity z = true <-> 0 <= z) -> 0 <= f (dicho_up x y (fun z : R => cond_positivity (f z)) n)f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n0 : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n0:nat -> RWn:=fun n0 : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n0:nat -> RH5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0n:natforall z : R, cond_positivity z = true <-> 0 <= zf:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0forall n : nat, f (Vn n) <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n0 : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n0:nat -> RWn:=fun n0 : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n0:nat -> RH5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0n:natH7:forall z : R, cond_positivity z = true <-> 0 <= z0 <= f (dicho_up x y (fun z : R => cond_positivity (f z)) n)f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n0 : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n0:nat -> RWn:=fun n0 : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n0:nat -> RH5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0n:natforall z : R, cond_positivity z = true <-> 0 <= zf:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0forall n : nat, f (Vn n) <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n0 : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n0:nat -> RWn:=fun n0 : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n0:nat -> RH5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0n:natH7:forall z : R, cond_positivity z = true <-> 0 <= zH8:(fun z : R => cond_positivity (f z)) y = true -> (fun z : R => cond_positivity (f z)) (dicho_up x y (fun z : R => cond_positivity (f z)) n) = true0 <= f (dicho_up x y (fun z : R => cond_positivity (f z)) n)f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n0 : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n0:nat -> RWn:=fun n0 : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n0:nat -> RH5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0n:natforall z : R, cond_positivity z = true <-> 0 <= zf:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0forall n : nat, f (Vn n) <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n0 : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n0:nat -> RWn:=fun n0 : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n0:nat -> RH5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0n:natH7:forall z : R, cond_positivity z = true <-> 0 <= zH8:(fun z : R => cond_positivity (f z)) y = true -> (fun z : R => cond_positivity (f z)) (dicho_up x y (fun z : R => cond_positivity (f z)) n) = trueH9:cond_positivity (f (dicho_up x y (fun z : R => cond_positivity (f z)) n)) = true -> 0 <= f (dicho_up x y (fun z : R => cond_positivity (f z)) n)H10:0 <= f (dicho_up x y (fun z : R => cond_positivity (f z)) n) -> cond_positivity (f (dicho_up x y (fun z : R => cond_positivity (f z)) n)) = true0 <= f (dicho_up x y (fun z : R => cond_positivity (f z)) n)f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n0 : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n0:nat -> RWn:=fun n0 : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n0:nat -> RH5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0n:natforall z : R, cond_positivity z = true <-> 0 <= zf:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0forall n : nat, f (Vn n) <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n0 : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n0:nat -> RWn:=fun n0 : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n0:nat -> RH5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0n:natH7:forall z : R, cond_positivity z = true <-> 0 <= zH8:(fun z : R => cond_positivity (f z)) y = true -> (fun z : R => cond_positivity (f z)) (dicho_up x y (fun z : R => cond_positivity (f z)) n) = trueH9:cond_positivity (f (dicho_up x y (fun z : R => cond_positivity (f z)) n)) = true -> 0 <= f (dicho_up x y (fun z : R => cond_positivity (f z)) n)H10:0 <= f (dicho_up x y (fun z : R => cond_positivity (f z)) n) -> cond_positivity (f (dicho_up x y (fun z : R => cond_positivity (f z)) n)) = truecond_positivity (f (dicho_up x y (fun z : R => cond_positivity (f z)) n)) = truef:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n0 : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n0:nat -> RWn:=fun n0 : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n0:nat -> RH5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0n:natforall z : R, cond_positivity z = true <-> 0 <= zf:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0forall n : nat, f (Vn n) <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n0 : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n0:nat -> RWn:=fun n0 : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n0:nat -> RH5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0n:natH7:forall z : R, cond_positivity z = true <-> 0 <= zH8:(fun z : R => cond_positivity (f z)) y = true -> (fun z : R => cond_positivity (f z)) (dicho_up x y (fun z : R => cond_positivity (f z)) n) = trueH9:cond_positivity (f (dicho_up x y (fun z : R => cond_positivity (f z)) n)) = true -> 0 <= f (dicho_up x y (fun z : R => cond_positivity (f z)) n)H10:0 <= f (dicho_up x y (fun z : R => cond_positivity (f z)) n) -> cond_positivity (f (dicho_up x y (fun z : R => cond_positivity (f z)) n)) = truecond_positivity (f y) = truef:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n0 : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n0:nat -> RWn:=fun n0 : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n0:nat -> RH5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0n:natforall z : R, cond_positivity z = true <-> 0 <= zf:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0forall n : nat, f (Vn n) <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n0 : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n0:nat -> RWn:=fun n0 : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n0:nat -> RH5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0n:natH7:forall z : R, cond_positivity z = true <-> 0 <= zH8:(fun z : R => cond_positivity (f z)) y = true -> (fun z : R => cond_positivity (f z)) (dicho_up x y (fun z : R => cond_positivity (f z)) n) = trueH9:cond_positivity (f (dicho_up x y (fun z : R => cond_positivity (f z)) n)) = true -> 0 <= f (dicho_up x y (fun z : R => cond_positivity (f z)) n)H10:0 <= f (dicho_up x y (fun z : R => cond_positivity (f z)) n) -> cond_positivity (f (dicho_up x y (fun z : R => cond_positivity (f z)) n)) = trueH11:cond_positivity (f y) = true -> 0 <= f yH12:0 <= f y -> cond_positivity (f y) = truecond_positivity (f y) = truef:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n0 : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n0:nat -> RWn:=fun n0 : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n0:nat -> RH5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0n:natforall z : R, cond_positivity z = true <-> 0 <= zf:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0forall n : nat, f (Vn n) <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n0 : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n0:nat -> RWn:=fun n0 : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n0:nat -> RH5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0n:natH7:forall z : R, cond_positivity z = true <-> 0 <= zH8:(fun z : R => cond_positivity (f z)) y = true -> (fun z : R => cond_positivity (f z)) (dicho_up x y (fun z : R => cond_positivity (f z)) n) = trueH9:cond_positivity (f (dicho_up x y (fun z : R => cond_positivity (f z)) n)) = true -> 0 <= f (dicho_up x y (fun z : R => cond_positivity (f z)) n)H10:0 <= f (dicho_up x y (fun z : R => cond_positivity (f z)) n) -> cond_positivity (f (dicho_up x y (fun z : R => cond_positivity (f z)) n)) = trueH11:cond_positivity (f y) = true -> 0 <= f yH12:0 <= f y -> cond_positivity (f y) = true0 <= f yf:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n0 : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n0:nat -> RWn:=fun n0 : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n0:nat -> RH5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0n:natforall z : R, cond_positivity z = true <-> 0 <= zf:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0forall n : nat, f (Vn n) <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n0 : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n0:nat -> RWn:=fun n0 : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n0:nat -> RH5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0n:natforall z : R, cond_positivity z = true <-> 0 <= zf:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0forall n : nat, f (Vn n) <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z0 : R => cond_positivity (f z0))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z0 : R => cond_positivity (f z0))) l}x0:Rp:Un_cv (dicho_up x y (fun z0 : R => cond_positivity (f z0))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z0 : R => cond_positivity (f z0))) x0H4:x1 = x0Vn:=fun n0 : nat => dicho_lb x y (fun z0 : R => cond_positivity (f z0)) n0:nat -> RWn:=fun n0 : nat => dicho_up x y (fun z0 : R => cond_positivity (f z0)) n0:nat -> RH5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0n:natz:Rcond_positivity z = true <-> 0 <= zf:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0forall n : nat, f (Vn n) <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z0 : R => cond_positivity (f z0))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z0 : R => cond_positivity (f z0))) l}x0:Rp:Un_cv (dicho_up x y (fun z0 : R => cond_positivity (f z0))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z0 : R => cond_positivity (f z0))) x0H4:x1 = x0Vn:=fun n0 : nat => dicho_lb x y (fun z0 : R => cond_positivity (f z0)) n0:nat -> RWn:=fun n0 : nat => dicho_up x y (fun z0 : R => cond_positivity (f z0)) n0:nat -> RH5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0n:natz:R(if Rle_dec 0 z then true else false) = true <-> 0 <= zf:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0forall n : nat, f (Vn n) <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z0 : R => cond_positivity (f z0))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z0 : R => cond_positivity (f z0))) l}x0:Rp:Un_cv (dicho_up x y (fun z0 : R => cond_positivity (f z0))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z0 : R => cond_positivity (f z0))) x0H4:x1 = x0Vn:=fun n0 : nat => dicho_lb x y (fun z0 : R => cond_positivity (f z0)) n0:nat -> RWn:=fun n0 : nat => dicho_up x y (fun z0 : R => cond_positivity (f z0)) n0:nat -> RH5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0n:natz:Rr:0 <= ztrue = true <-> 0 <= zf:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z0 : R => cond_positivity (f z0))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z0 : R => cond_positivity (f z0))) l}x0:Rp:Un_cv (dicho_up x y (fun z0 : R => cond_positivity (f z0))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z0 : R => cond_positivity (f z0))) x0H4:x1 = x0Vn:=fun n0 : nat => dicho_lb x y (fun z0 : R => cond_positivity (f z0)) n0:nat -> RWn:=fun n0 : nat => dicho_up x y (fun z0 : R => cond_positivity (f z0)) n0:nat -> RH5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0n:natz:RHnotle:~ 0 <= zfalse = true <-> 0 <= zf:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0forall n : nat, f (Vn n) <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z0 : R => cond_positivity (f z0))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z0 : R => cond_positivity (f z0))) l}x0:Rp:Un_cv (dicho_up x y (fun z0 : R => cond_positivity (f z0))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z0 : R => cond_positivity (f z0))) x0H4:x1 = x0Vn:=fun n0 : nat => dicho_lb x y (fun z0 : R => cond_positivity (f z0)) n0:nat -> RWn:=fun n0 : nat => dicho_up x y (fun z0 : R => cond_positivity (f z0)) n0:nat -> RH5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0n:natz:Rr:0 <= ztrue = true -> 0 <= zf:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z0 : R => cond_positivity (f z0))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z0 : R => cond_positivity (f z0))) l}x0:Rp:Un_cv (dicho_up x y (fun z0 : R => cond_positivity (f z0))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z0 : R => cond_positivity (f z0))) x0H4:x1 = x0Vn:=fun n0 : nat => dicho_lb x y (fun z0 : R => cond_positivity (f z0)) n0:nat -> RWn:=fun n0 : nat => dicho_up x y (fun z0 : R => cond_positivity (f z0)) n0:nat -> RH5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0n:natz:Rr:0 <= z0 <= z -> true = truef:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z0 : R => cond_positivity (f z0))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z0 : R => cond_positivity (f z0))) l}x0:Rp:Un_cv (dicho_up x y (fun z0 : R => cond_positivity (f z0))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z0 : R => cond_positivity (f z0))) x0H4:x1 = x0Vn:=fun n0 : nat => dicho_lb x y (fun z0 : R => cond_positivity (f z0)) n0:nat -> RWn:=fun n0 : nat => dicho_up x y (fun z0 : R => cond_positivity (f z0)) n0:nat -> RH5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0n:natz:RHnotle:~ 0 <= zfalse = true <-> 0 <= zf:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0forall n : nat, f (Vn n) <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z0 : R => cond_positivity (f z0))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z0 : R => cond_positivity (f z0))) l}x0:Rp:Un_cv (dicho_up x y (fun z0 : R => cond_positivity (f z0))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z0 : R => cond_positivity (f z0))) x0H4:x1 = x0Vn:=fun n0 : nat => dicho_lb x y (fun z0 : R => cond_positivity (f z0)) n0:nat -> RWn:=fun n0 : nat => dicho_up x y (fun z0 : R => cond_positivity (f z0)) n0:nat -> RH5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0n:natz:Rr:0 <= z0 <= z -> true = truef:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z0 : R => cond_positivity (f z0))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z0 : R => cond_positivity (f z0))) l}x0:Rp:Un_cv (dicho_up x y (fun z0 : R => cond_positivity (f z0))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z0 : R => cond_positivity (f z0))) x0H4:x1 = x0Vn:=fun n0 : nat => dicho_lb x y (fun z0 : R => cond_positivity (f z0)) n0:nat -> RWn:=fun n0 : nat => dicho_up x y (fun z0 : R => cond_positivity (f z0)) n0:nat -> RH5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0n:natz:RHnotle:~ 0 <= zfalse = true <-> 0 <= zf:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0forall n : nat, f (Vn n) <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z0 : R => cond_positivity (f z0))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z0 : R => cond_positivity (f z0))) l}x0:Rp:Un_cv (dicho_up x y (fun z0 : R => cond_positivity (f z0))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z0 : R => cond_positivity (f z0))) x0H4:x1 = x0Vn:=fun n0 : nat => dicho_lb x y (fun z0 : R => cond_positivity (f z0)) n0:nat -> RWn:=fun n0 : nat => dicho_up x y (fun z0 : R => cond_positivity (f z0)) n0:nat -> RH5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0n:natz:RHnotle:~ 0 <= zfalse = true <-> 0 <= zf:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0forall n : nat, f (Vn n) <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z0 : R => cond_positivity (f z0))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z0 : R => cond_positivity (f z0))) l}x0:Rp:Un_cv (dicho_up x y (fun z0 : R => cond_positivity (f z0))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z0 : R => cond_positivity (f z0))) x0H4:x1 = x0Vn:=fun n0 : nat => dicho_lb x y (fun z0 : R => cond_positivity (f z0)) n0:nat -> RWn:=fun n0 : nat => dicho_up x y (fun z0 : R => cond_positivity (f z0)) n0:nat -> RH5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0n:natz:RHnotle:~ 0 <= zfalse = true -> 0 <= zf:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z0 : R => cond_positivity (f z0))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z0 : R => cond_positivity (f z0))) l}x0:Rp:Un_cv (dicho_up x y (fun z0 : R => cond_positivity (f z0))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z0 : R => cond_positivity (f z0))) x0H4:x1 = x0Vn:=fun n0 : nat => dicho_lb x y (fun z0 : R => cond_positivity (f z0)) n0:nat -> RWn:=fun n0 : nat => dicho_up x y (fun z0 : R => cond_positivity (f z0)) n0:nat -> RH5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0n:natz:RHnotle:~ 0 <= z0 <= z -> false = truef:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0forall n : nat, f (Vn n) <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z0 : R => cond_positivity (f z0))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z0 : R => cond_positivity (f z0))) l}x0:Rp:Un_cv (dicho_up x y (fun z0 : R => cond_positivity (f z0))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z0 : R => cond_positivity (f z0))) x0H4:x1 = x0Vn:=fun n0 : nat => dicho_lb x y (fun z0 : R => cond_positivity (f z0)) n0:nat -> RWn:=fun n0 : nat => dicho_up x y (fun z0 : R => cond_positivity (f z0)) n0:nat -> RH5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0n:natz:RHnotle:~ 0 <= z0 <= z -> false = truef:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0forall n : nat, f (Vn n) <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z0 : R => cond_positivity (f z0))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z0 : R => cond_positivity (f z0))) l}x0:Rp:Un_cv (dicho_up x y (fun z0 : R => cond_positivity (f z0))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z0 : R => cond_positivity (f z0))) x0H4:x1 = x0Vn:=fun n0 : nat => dicho_lb x y (fun z0 : R => cond_positivity (f z0)) n0:nat -> RWn:=fun n0 : nat => dicho_up x y (fun z0 : R => cond_positivity (f z0)) n0:nat -> RH5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0n:natz:RHnotle:~ 0 <= zH7:0 <= zfalse = truef:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0forall n : nat, f (Vn n) <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0forall n : nat, f (Vn n) <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0forall n : nat, f (dicho_lb x y (fun z : R => cond_positivity (f z)) n) <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0(forall z : R, cond_positivity z = false <-> z < 0) -> forall n : nat, f (dicho_lb x y (fun z : R => cond_positivity (f z)) n) <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0forall z : R, cond_positivity z = false <-> z < 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n0 : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n0:nat -> RWn:=fun n0 : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n0:nat -> RH5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0H7:forall z : R, cond_positivity z = false <-> z < 0n:natf (dicho_lb x y (fun z : R => cond_positivity (f z)) n) <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0forall z : R, cond_positivity z = false <-> z < 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n0 : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n0:nat -> RWn:=fun n0 : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n0:nat -> RH5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0H7:forall z : R, cond_positivity z = false <-> z < 0n:natH8:(fun z : R => cond_positivity (f z)) x = false -> (fun z : R => cond_positivity (f z)) (dicho_lb x y (fun z : R => cond_positivity (f z)) n) = falsef (dicho_lb x y (fun z : R => cond_positivity (f z)) n) <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0forall z : R, cond_positivity z = false <-> z < 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n0 : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n0:nat -> RWn:=fun n0 : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n0:nat -> RH5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0H7:forall z : R, cond_positivity z = false <-> z < 0n:natH8:(fun z : R => cond_positivity (f z)) x = false -> (fun z : R => cond_positivity (f z)) (dicho_lb x y (fun z : R => cond_positivity (f z)) n) = falsef (dicho_lb x y (fun z : R => cond_positivity (f z)) n) < 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0forall z : R, cond_positivity z = false <-> z < 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n0 : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n0:nat -> RWn:=fun n0 : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n0:nat -> RH5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0H7:forall z : R, cond_positivity z = false <-> z < 0n:natH8:(fun z : R => cond_positivity (f z)) x = false -> (fun z : R => cond_positivity (f z)) (dicho_lb x y (fun z : R => cond_positivity (f z)) n) = falseH9:cond_positivity (f (dicho_lb x y (fun z : R => cond_positivity (f z)) n)) = false -> f (dicho_lb x y (fun z : R => cond_positivity (f z)) n) < 0H10:f (dicho_lb x y (fun z : R => cond_positivity (f z)) n) < 0 -> cond_positivity (f (dicho_lb x y (fun z : R => cond_positivity (f z)) n)) = falsef (dicho_lb x y (fun z : R => cond_positivity (f z)) n) < 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0forall z : R, cond_positivity z = false <-> z < 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n0 : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n0:nat -> RWn:=fun n0 : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n0:nat -> RH5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0H7:forall z : R, cond_positivity z = false <-> z < 0n:natH8:(fun z : R => cond_positivity (f z)) x = false -> (fun z : R => cond_positivity (f z)) (dicho_lb x y (fun z : R => cond_positivity (f z)) n) = falseH9:cond_positivity (f (dicho_lb x y (fun z : R => cond_positivity (f z)) n)) = false -> f (dicho_lb x y (fun z : R => cond_positivity (f z)) n) < 0H10:f (dicho_lb x y (fun z : R => cond_positivity (f z)) n) < 0 -> cond_positivity (f (dicho_lb x y (fun z : R => cond_positivity (f z)) n)) = falsecond_positivity (f (dicho_lb x y (fun z : R => cond_positivity (f z)) n)) = falsef:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0forall z : R, cond_positivity z = false <-> z < 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n0 : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n0:nat -> RWn:=fun n0 : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n0:nat -> RH5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0H7:forall z : R, cond_positivity z = false <-> z < 0n:natH8:(fun z : R => cond_positivity (f z)) x = false -> (fun z : R => cond_positivity (f z)) (dicho_lb x y (fun z : R => cond_positivity (f z)) n) = falseH9:cond_positivity (f (dicho_lb x y (fun z : R => cond_positivity (f z)) n)) = false -> f (dicho_lb x y (fun z : R => cond_positivity (f z)) n) < 0H10:f (dicho_lb x y (fun z : R => cond_positivity (f z)) n) < 0 -> cond_positivity (f (dicho_lb x y (fun z : R => cond_positivity (f z)) n)) = falsecond_positivity (f x) = falsef:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0forall z : R, cond_positivity z = false <-> z < 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n0 : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n0:nat -> RWn:=fun n0 : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n0:nat -> RH5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0H7:forall z : R, cond_positivity z = false <-> z < 0n:natH8:(fun z : R => cond_positivity (f z)) x = false -> (fun z : R => cond_positivity (f z)) (dicho_lb x y (fun z : R => cond_positivity (f z)) n) = falseH9:cond_positivity (f (dicho_lb x y (fun z : R => cond_positivity (f z)) n)) = false -> f (dicho_lb x y (fun z : R => cond_positivity (f z)) n) < 0H10:f (dicho_lb x y (fun z : R => cond_positivity (f z)) n) < 0 -> cond_positivity (f (dicho_lb x y (fun z : R => cond_positivity (f z)) n)) = falseH11:cond_positivity (f x) = false -> f x < 0H12:f x < 0 -> cond_positivity (f x) = falsecond_positivity (f x) = falsef:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0forall z : R, cond_positivity z = false <-> z < 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n0 : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n0:nat -> RWn:=fun n0 : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n0:nat -> RH5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0H7:forall z : R, cond_positivity z = false <-> z < 0n:natH8:(fun z : R => cond_positivity (f z)) x = false -> (fun z : R => cond_positivity (f z)) (dicho_lb x y (fun z : R => cond_positivity (f z)) n) = falseH9:cond_positivity (f (dicho_lb x y (fun z : R => cond_positivity (f z)) n)) = false -> f (dicho_lb x y (fun z : R => cond_positivity (f z)) n) < 0H10:f (dicho_lb x y (fun z : R => cond_positivity (f z)) n) < 0 -> cond_positivity (f (dicho_lb x y (fun z : R => cond_positivity (f z)) n)) = falseH11:cond_positivity (f x) = false -> f x < 0H12:f x < 0 -> cond_positivity (f x) = falsef x < 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0forall z : R, cond_positivity z = false <-> z < 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0forall z : R, cond_positivity z = false <-> z < 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z0 : R => cond_positivity (f z0))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z0 : R => cond_positivity (f z0))) l}x0:Rp:Un_cv (dicho_up x y (fun z0 : R => cond_positivity (f z0))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z0 : R => cond_positivity (f z0))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z0 : R => cond_positivity (f z0)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z0 : R => cond_positivity (f z0)) n:nat -> RH5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0z:Rcond_positivity z = false <-> z < 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z0 : R => cond_positivity (f z0))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z0 : R => cond_positivity (f z0))) l}x0:Rp:Un_cv (dicho_up x y (fun z0 : R => cond_positivity (f z0))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z0 : R => cond_positivity (f z0))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z0 : R => cond_positivity (f z0)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z0 : R => cond_positivity (f z0)) n:nat -> RH5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0z:R(if Rle_dec 0 z then true else false) = false <-> z < 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z0 : R => cond_positivity (f z0))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z0 : R => cond_positivity (f z0))) l}x0:Rp:Un_cv (dicho_up x y (fun z0 : R => cond_positivity (f z0))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z0 : R => cond_positivity (f z0))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z0 : R => cond_positivity (f z0)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z0 : R => cond_positivity (f z0)) n:nat -> RH5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0z:RHle:0 <= ztrue = false <-> z < 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z0 : R => cond_positivity (f z0))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z0 : R => cond_positivity (f z0))) l}x0:Rp:Un_cv (dicho_up x y (fun z0 : R => cond_positivity (f z0))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z0 : R => cond_positivity (f z0))) x0H4:x1 = x0Vn:=fun n0 : nat => dicho_lb x y (fun z0 : R => cond_positivity (f z0)) n0:nat -> RWn:=fun n0 : nat => dicho_up x y (fun z0 : R => cond_positivity (f z0)) n0:nat -> RH5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0z:Rn:~ 0 <= zfalse = false <-> z < 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z0 : R => cond_positivity (f z0))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z0 : R => cond_positivity (f z0))) l}x0:Rp:Un_cv (dicho_up x y (fun z0 : R => cond_positivity (f z0))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z0 : R => cond_positivity (f z0))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z0 : R => cond_positivity (f z0)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z0 : R => cond_positivity (f z0)) n:nat -> RH5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0z:RHle:0 <= ztrue = false -> z < 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z0 : R => cond_positivity (f z0))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z0 : R => cond_positivity (f z0))) l}x0:Rp:Un_cv (dicho_up x y (fun z0 : R => cond_positivity (f z0))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z0 : R => cond_positivity (f z0))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z0 : R => cond_positivity (f z0)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z0 : R => cond_positivity (f z0)) n:nat -> RH5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0z:RHle:0 <= zz < 0 -> true = falsef:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z0 : R => cond_positivity (f z0))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z0 : R => cond_positivity (f z0))) l}x0:Rp:Un_cv (dicho_up x y (fun z0 : R => cond_positivity (f z0))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z0 : R => cond_positivity (f z0))) x0H4:x1 = x0Vn:=fun n0 : nat => dicho_lb x y (fun z0 : R => cond_positivity (f z0)) n0:nat -> RWn:=fun n0 : nat => dicho_up x y (fun z0 : R => cond_positivity (f z0)) n0:nat -> RH5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0z:Rn:~ 0 <= zfalse = false <-> z < 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z0 : R => cond_positivity (f z0))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z0 : R => cond_positivity (f z0))) l}x0:Rp:Un_cv (dicho_up x y (fun z0 : R => cond_positivity (f z0))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z0 : R => cond_positivity (f z0))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z0 : R => cond_positivity (f z0)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z0 : R => cond_positivity (f z0)) n:nat -> RH5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0z:RHle:0 <= zz < 0 -> true = falsef:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z0 : R => cond_positivity (f z0))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z0 : R => cond_positivity (f z0))) l}x0:Rp:Un_cv (dicho_up x y (fun z0 : R => cond_positivity (f z0))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z0 : R => cond_positivity (f z0))) x0H4:x1 = x0Vn:=fun n0 : nat => dicho_lb x y (fun z0 : R => cond_positivity (f z0)) n0:nat -> RWn:=fun n0 : nat => dicho_up x y (fun z0 : R => cond_positivity (f z0)) n0:nat -> RH5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0z:Rn:~ 0 <= zfalse = false <-> z < 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z0 : R => cond_positivity (f z0))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z0 : R => cond_positivity (f z0))) l}x0:Rp:Un_cv (dicho_up x y (fun z0 : R => cond_positivity (f z0))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z0 : R => cond_positivity (f z0))) x0H4:x1 = x0Vn:=fun n0 : nat => dicho_lb x y (fun z0 : R => cond_positivity (f z0)) n0:nat -> RWn:=fun n0 : nat => dicho_up x y (fun z0 : R => cond_positivity (f z0)) n0:nat -> RH5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0z:Rn:~ 0 <= zfalse = false <-> z < 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z0 : R => cond_positivity (f z0))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z0 : R => cond_positivity (f z0))) l}x0:Rp:Un_cv (dicho_up x y (fun z0 : R => cond_positivity (f z0))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z0 : R => cond_positivity (f z0))) x0H4:x1 = x0Vn:=fun n0 : nat => dicho_lb x y (fun z0 : R => cond_positivity (f z0)) n0:nat -> RWn:=fun n0 : nat => dicho_up x y (fun z0 : R => cond_positivity (f z0)) n0:nat -> RH5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0z:Rn:~ 0 <= zfalse = false -> z < 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z0 : R => cond_positivity (f z0))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z0 : R => cond_positivity (f z0))) l}x0:Rp:Un_cv (dicho_up x y (fun z0 : R => cond_positivity (f z0))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z0 : R => cond_positivity (f z0))) x0H4:x1 = x0Vn:=fun n0 : nat => dicho_lb x y (fun z0 : R => cond_positivity (f z0)) n0:nat -> RWn:=fun n0 : nat => dicho_up x y (fun z0 : R => cond_positivity (f z0)) n0:nat -> RH5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0z:Rn:~ 0 <= zz < 0 -> false = falsef:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z0 : R => cond_positivity (f z0))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z0 : R => cond_positivity (f z0))) l}x0:Rp:Un_cv (dicho_up x y (fun z0 : R => cond_positivity (f z0))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z0 : R => cond_positivity (f z0))) x0H4:x1 = x0Vn:=fun n0 : nat => dicho_lb x y (fun z0 : R => cond_positivity (f z0)) n0:nat -> RWn:=fun n0 : nat => dicho_up x y (fun z0 : R => cond_positivity (f z0)) n0:nat -> RH5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0z:Rn:~ 0 <= zz < 0 -> false = falsef:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RUn_cv Wn x0 -> (forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RUn_cv Wn x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Wn x0H6:forall n : nat, 0 <= f (Wn n)0 <= f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RUn_cv Wn x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Wn x0H6:forall n : nat, 0 <= f (Wn n)x <= x0 <= yf:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Wn x0H6:forall n : nat, 0 <= f (Wn n)Temp:x <= x0 <= y0 <= f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RUn_cv Wn x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Wn x0H6:forall n : nat, 0 <= f (Wn n)Temp:x <= x0 <= y0 <= f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RUn_cv Wn x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Wn x0H6:forall n : nat, 0 <= f (Wn n)Temp:x <= x0 <= yH7:Un_cv (fun i : nat => f (Wn i)) (f x0)0 <= f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RUn_cv Wn x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Wn x0H6:forall n : nat, 0 <= f (Wn n)Temp:x <= x0 <= yH7:Un_cv (fun i : nat => f (Wn i)) (f x0)Hlt:0 < f x00 <= f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Wn x0H6:forall n : nat, 0 <= f (Wn n)Temp:x <= x0 <= yH7:Un_cv (fun i : nat => f (Wn i)) (f x0)0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Wn x0H6:forall n : nat, 0 <= f (Wn n)Temp:x <= x0 <= yH7:Un_cv (fun i : nat => f (Wn i)) (f x0)Hgt:0 > f x00 <= f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RUn_cv Wn x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Wn x0H6:forall n : nat, 0 <= f (Wn n)Temp:x <= x0 <= yH7:Un_cv (fun i : nat => f (Wn i)) (f x0)0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Wn x0H6:forall n : nat, 0 <= f (Wn n)Temp:x <= x0 <= yH7:Un_cv (fun i : nat => f (Wn i)) (f x0)Hgt:0 > f x00 <= f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RUn_cv Wn x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Wn x0H6:forall n : nat, 0 <= f (Wn n)Temp:x <= x0 <= yH7:Un_cv (fun i : nat => f (Wn i)) (f x0)Hgt:0 > f x00 <= f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RUn_cv Wn x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Wn x0H6:forall n : nat, 0 <= f (Wn n)Temp:x <= x0 <= yH7:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> Rabs (f (Wn n) - f x0) < epsHgt:0 > f x00 <= f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RUn_cv Wn x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Wn x0H6:forall n : nat, 0 <= f (Wn n)Temp:x <= x0 <= yH7:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> Rabs (f (Wn n) - f x0) < epsHgt:0 > f x00 < - f x0 -> 0 <= f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Wn x0H6:forall n : nat, 0 <= f (Wn n)Temp:x <= x0 <= yH7:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> Rabs (f (Wn n) - f x0) < epsHgt:0 > f x00 < - f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RUn_cv Wn x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Wn x0H6:forall n : nat, 0 <= f (Wn n)Temp:x <= x0 <= yH7:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> Rabs (f (Wn n) - f x0) < epsHgt:0 > f x0H8:0 < - f x00 <= f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Wn x0H6:forall n : nat, 0 <= f (Wn n)Temp:x <= x0 <= yH7:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> Rabs (f (Wn n) - f x0) < epsHgt:0 > f x00 < - f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RUn_cv Wn x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Wn x0H6:forall n : nat, 0 <= f (Wn n)Temp:x <= x0 <= yH7:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> Rabs (f (Wn n) - f x0) < epsHgt:0 > f x0H8:0 < - f x0x2:natH9:forall n : nat, (n >= x2)%nat -> Rabs (f (Wn n) - f x0) < - f x00 <= f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Wn x0H6:forall n : nat, 0 <= f (Wn n)Temp:x <= x0 <= yH7:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> Rabs (f (Wn n) - f x0) < epsHgt:0 > f x00 < - f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RUn_cv Wn x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Wn x0H6:forall n : nat, 0 <= f (Wn n)Temp:x <= x0 <= yH7:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> Rabs (f (Wn n) - f x0) < epsHgt:0 > f x0H8:0 < - f x0x2:natH9:forall n : nat, (n >= x2)%nat -> Rabs (f (Wn n) - f x0) < - f x0H10:(x2 >= x2)%nat0 <= f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Wn x0H6:forall n : nat, 0 <= f (Wn n)Temp:x <= x0 <= yH7:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> Rabs (f (Wn n) - f x0) < epsHgt:0 > f x00 < - f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RUn_cv Wn x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Wn x0H6:forall n : nat, 0 <= f (Wn n)Temp:x <= x0 <= yH7:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> Rabs (f (Wn n) - f x0) < epsHgt:0 > f x0H8:0 < - f x0x2:natH9:forall n : nat, (n >= x2)%nat -> Rabs (f (Wn n) - f x0) < - f x0H10:(x2 >= x2)%natH11:Rabs (f (Wn x2) - f x0) < - f x00 <= f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Wn x0H6:forall n : nat, 0 <= f (Wn n)Temp:x <= x0 <= yH7:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> Rabs (f (Wn n) - f x0) < epsHgt:0 > f x00 < - f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RUn_cv Wn x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Wn x0H6:forall n : nat, 0 <= f (Wn n)Temp:x <= x0 <= yH7:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> Rabs (f (Wn n) - f x0) < epsHgt:0 > f x0H8:0 < - f x0x2:natH9:forall n : nat, (n >= x2)%nat -> Rabs (f (Wn n) - f x0) < - f x0H10:(x2 >= x2)%natH11:f (Wn x2) - f x0 < - f x00 <= f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Wn x0H6:forall n : nat, 0 <= f (Wn n)Temp:x <= x0 <= yH7:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> Rabs (f (Wn n) - f x0) < epsHgt:0 > f x0H8:0 < - f x0x2:natH9:forall n : nat, (n >= x2)%nat -> Rabs (f (Wn n) - f x0) < - f x0H10:(x2 >= x2)%natH11:Rabs (f (Wn x2) - f x0) < - f x0f (Wn x2) - f x0 >= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Wn x0H6:forall n : nat, 0 <= f (Wn n)Temp:x <= x0 <= yH7:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> Rabs (f (Wn n) - f x0) < epsHgt:0 > f x00 < - f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RUn_cv Wn x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Wn x0H6:forall n : nat, 0 <= f (Wn n)Temp:x <= x0 <= yH7:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> Rabs (f (Wn n) - f x0) < epsHgt:0 > f x0H8:0 < - f x0x2:natH9:forall n : nat, (n >= x2)%nat -> Rabs (f (Wn n) - f x0) < - f x0H10:(x2 >= x2)%natH11:f (Wn x2) - f x0 < - f x0 + 00 <= f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Wn x0H6:forall n : nat, 0 <= f (Wn n)Temp:x <= x0 <= yH7:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> Rabs (f (Wn n) - f x0) < epsHgt:0 > f x0H8:0 < - f x0x2:natH9:forall n : nat, (n >= x2)%nat -> Rabs (f (Wn n) - f x0) < - f x0H10:(x2 >= x2)%natH11:Rabs (f (Wn x2) - f x0) < - f x0f (Wn x2) - f x0 >= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Wn x0H6:forall n : nat, 0 <= f (Wn n)Temp:x <= x0 <= yH7:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> Rabs (f (Wn n) - f x0) < epsHgt:0 > f x00 < - f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RUn_cv Wn x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Wn x0H6:forall n : nat, 0 <= f (Wn n)Temp:x <= x0 <= yH7:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> Rabs (f (Wn n) - f x0) < epsHgt:0 > f x0H8:0 < - f x0x2:natH9:forall n : nat, (n >= x2)%nat -> Rabs (f (Wn n) - f x0) < - f x0H10:(x2 >= x2)%natH11:- f x0 + f (Wn x2) < - f x0 + 00 <= f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Wn x0H6:forall n : nat, 0 <= f (Wn n)Temp:x <= x0 <= yH7:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> Rabs (f (Wn n) - f x0) < epsHgt:0 > f x0H8:0 < - f x0x2:natH9:forall n : nat, (n >= x2)%nat -> Rabs (f (Wn n) - f x0) < - f x0H10:(x2 >= x2)%natH11:Rabs (f (Wn x2) - f x0) < - f x0f (Wn x2) - f x0 >= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Wn x0H6:forall n : nat, 0 <= f (Wn n)Temp:x <= x0 <= yH7:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> Rabs (f (Wn n) - f x0) < epsHgt:0 > f x00 < - f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RUn_cv Wn x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Wn x0H6:forall n : nat, 0 <= f (Wn n)Temp:x <= x0 <= yH7:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> Rabs (f (Wn n) - f x0) < epsHgt:0 > f x0H8:0 < - f x0x2:natH9:forall n : nat, (n >= x2)%nat -> Rabs (f (Wn n) - f x0) < - f x0H10:(x2 >= x2)%natH11:- f x0 + f (Wn x2) < - f x0 + 0H12:f (Wn x2) < 00 <= f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Wn x0H6:forall n : nat, 0 <= f (Wn n)Temp:x <= x0 <= yH7:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> Rabs (f (Wn n) - f x0) < epsHgt:0 > f x0H8:0 < - f x0x2:natH9:forall n : nat, (n >= x2)%nat -> Rabs (f (Wn n) - f x0) < - f x0H10:(x2 >= x2)%natH11:Rabs (f (Wn x2) - f x0) < - f x0f (Wn x2) - f x0 >= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Wn x0H6:forall n : nat, 0 <= f (Wn n)Temp:x <= x0 <= yH7:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> Rabs (f (Wn n) - f x0) < epsHgt:0 > f x00 < - f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RUn_cv Wn x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Wn x0H6:forall n : nat, 0 <= f (Wn n)Temp:x <= x0 <= yH7:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> Rabs (f (Wn n) - f x0) < epsHgt:0 > f x0H8:0 < - f x0x2:natH9:forall n : nat, (n >= x2)%nat -> Rabs (f (Wn n) - f x0) < - f x0H10:(x2 >= x2)%natH11:- f x0 + f (Wn x2) < - f x0 + 0H12:f (Wn x2) < 0H13:0 <= f (Wn x2)0 <= f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Wn x0H6:forall n : nat, 0 <= f (Wn n)Temp:x <= x0 <= yH7:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> Rabs (f (Wn n) - f x0) < epsHgt:0 > f x0H8:0 < - f x0x2:natH9:forall n : nat, (n >= x2)%nat -> Rabs (f (Wn n) - f x0) < - f x0H10:(x2 >= x2)%natH11:Rabs (f (Wn x2) - f x0) < - f x0f (Wn x2) - f x0 >= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Wn x0H6:forall n : nat, 0 <= f (Wn n)Temp:x <= x0 <= yH7:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> Rabs (f (Wn n) - f x0) < epsHgt:0 > f x00 < - f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RUn_cv Wn x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Wn x0H6:forall n : nat, 0 <= f (Wn n)Temp:x <= x0 <= yH7:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> Rabs (f (Wn n) - f x0) < epsHgt:0 > f x0H8:0 < - f x0x2:natH9:forall n : nat, (n >= x2)%nat -> Rabs (f (Wn n) - f x0) < - f x0H10:(x2 >= x2)%natH11:Rabs (f (Wn x2) - f x0) < - f x0f (Wn x2) - f x0 >= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Wn x0H6:forall n : nat, 0 <= f (Wn n)Temp:x <= x0 <= yH7:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> Rabs (f (Wn n) - f x0) < epsHgt:0 > f x00 < - f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RUn_cv Wn x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Wn x0H6:forall n : nat, 0 <= f (Wn n)Temp:x <= x0 <= yH7:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> Rabs (f (Wn n) - f x0) < epsHgt:0 > f x0H8:0 < - f x0x2:natH9:forall n : nat, (n >= x2)%nat -> Rabs (f (Wn n) - f x0) < - f x0H10:(x2 >= x2)%natH11:Rabs (f (Wn x2) - f x0) < - f x00 <= f (Wn x2)f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Wn x0H6:forall n : nat, 0 <= f (Wn n)Temp:x <= x0 <= yH7:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> Rabs (f (Wn n) - f x0) < epsHgt:0 > f x0H8:0 < - f x0x2:natH9:forall n : nat, (n >= x2)%nat -> Rabs (f (Wn n) - f x0) < - f x0H10:(x2 >= x2)%natH11:Rabs (f (Wn x2) - f x0) < - f x00 < - f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Wn x0H6:forall n : nat, 0 <= f (Wn n)Temp:x <= x0 <= yH7:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> Rabs (f (Wn n) - f x0) < epsHgt:0 > f x00 < - f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RUn_cv Wn x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Wn x0H6:forall n : nat, 0 <= f (Wn n)Temp:x <= x0 <= yH7:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> Rabs (f (Wn n) - f x0) < epsHgt:0 > f x0H8:0 < - f x0x2:natH9:forall n : nat, (n >= x2)%nat -> Rabs (f (Wn n) - f x0) < - f x0H10:(x2 >= x2)%natH11:Rabs (f (Wn x2) - f x0) < - f x00 < - f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Wn x0H6:forall n : nat, 0 <= f (Wn n)Temp:x <= x0 <= yH7:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> Rabs (f (Wn n) - f x0) < epsHgt:0 > f x00 < - f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RUn_cv Wn x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Wn x0H6:forall n : nat, 0 <= f (Wn n)Temp:x <= x0 <= yH7:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> Rabs (f (Wn n) - f x0) < epsHgt:0 > f x00 < - f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RUn_cv Wn x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RUn_cv Wn x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> R(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RUn_cv Vn x0 -> (forall n : nat, f (Vn n) <= 0) -> f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RUn_cv Vn x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Vn x0H6:forall n : nat, f (Vn n) <= 0f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RUn_cv Vn x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Vn x0H6:forall n : nat, f (Vn n) <= 0x <= x0 <= yf:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Vn x0H6:forall n : nat, f (Vn n) <= 0Temp:x <= x0 <= yf x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RUn_cv Vn x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Vn x0H6:forall n : nat, f (Vn n) <= 0Temp:x <= x0 <= yf x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RUn_cv Vn x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Vn x0H6:forall n : nat, f (Vn n) <= 0Temp:x <= x0 <= yH7:Un_cv (fun i : nat => f (Vn i)) (f x0)f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RUn_cv Vn x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Vn x0H6:forall n : nat, f (Vn n) <= 0Temp:x <= x0 <= yH7:Un_cv (fun i : nat => f (Vn i)) (f x0)Hlt:0 < f x0f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Vn x0H6:forall n : nat, f (Vn n) <= 0Temp:x <= x0 <= yH7:Un_cv (fun i : nat => f (Vn i)) (f x0)Heq:0 = f x0f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Vn x0H6:forall n : nat, f (Vn n) <= 0Temp:x <= x0 <= yH7:Un_cv (fun i : nat => f (Vn i)) (f x0)r:0 > f x0f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RUn_cv Vn x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Vn x0H6:forall n : nat, f (Vn n) <= 0Temp:x <= x0 <= yH7:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> Rabs (f (Vn n) - f x0) < epsHlt:0 < f x0f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Vn x0H6:forall n : nat, f (Vn n) <= 0Temp:x <= x0 <= yH7:Un_cv (fun i : nat => f (Vn i)) (f x0)Heq:0 = f x0f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Vn x0H6:forall n : nat, f (Vn n) <= 0Temp:x <= x0 <= yH7:Un_cv (fun i : nat => f (Vn i)) (f x0)r:0 > f x0f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RUn_cv Vn x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Vn x0H6:forall n : nat, f (Vn n) <= 0Temp:x <= x0 <= yH7:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> Rabs (f (Vn n) - f x0) < epsHlt:0 < f x0x2:natH8:forall n : nat, (n >= x2)%nat -> Rabs (f (Vn n) - f x0) < f x0f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Vn x0H6:forall n : nat, f (Vn n) <= 0Temp:x <= x0 <= yH7:Un_cv (fun i : nat => f (Vn i)) (f x0)Heq:0 = f x0f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Vn x0H6:forall n : nat, f (Vn n) <= 0Temp:x <= x0 <= yH7:Un_cv (fun i : nat => f (Vn i)) (f x0)r:0 > f x0f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RUn_cv Vn x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Vn x0H6:forall n : nat, f (Vn n) <= 0Temp:x <= x0 <= yH7:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> Rabs (f (Vn n) - f x0) < epsHlt:0 < f x0x2:natH8:forall n : nat, (n >= x2)%nat -> Rabs (f (Vn n) - f x0) < f x0H9:(x2 >= x2)%natf x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Vn x0H6:forall n : nat, f (Vn n) <= 0Temp:x <= x0 <= yH7:Un_cv (fun i : nat => f (Vn i)) (f x0)Heq:0 = f x0f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Vn x0H6:forall n : nat, f (Vn n) <= 0Temp:x <= x0 <= yH7:Un_cv (fun i : nat => f (Vn i)) (f x0)r:0 > f x0f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RUn_cv Vn x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Vn x0H6:forall n : nat, f (Vn n) <= 0Temp:x <= x0 <= yH7:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> Rabs (f (Vn n) - f x0) < epsHlt:0 < f x0x2:natH8:forall n : nat, (n >= x2)%nat -> Rabs (f (Vn n) - f x0) < f x0H9:(x2 >= x2)%natH10:Rabs (f (Vn x2) - f x0) < f x0f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Vn x0H6:forall n : nat, f (Vn n) <= 0Temp:x <= x0 <= yH7:Un_cv (fun i : nat => f (Vn i)) (f x0)Heq:0 = f x0f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Vn x0H6:forall n : nat, f (Vn n) <= 0Temp:x <= x0 <= yH7:Un_cv (fun i : nat => f (Vn i)) (f x0)r:0 > f x0f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RUn_cv Vn x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Vn x0H6:forall n : nat, f (Vn n) <= 0Temp:x <= x0 <= yH7:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> Rabs (f (Vn n) - f x0) < epsHlt:0 < f x0x2:natH8:forall n : nat, (n >= x2)%nat -> Rabs (f (Vn n) - f x0) < f x0H9:(x2 >= x2)%natH10:- (f (Vn x2) - f x0) < f x0f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Vn x0H6:forall n : nat, f (Vn n) <= 0Temp:x <= x0 <= yH7:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> Rabs (f (Vn n) - f x0) < epsHlt:0 < f x0x2:natH8:forall n : nat, (n >= x2)%nat -> Rabs (f (Vn n) - f x0) < f x0H9:(x2 >= x2)%natH10:Rabs (f (Vn x2) - f x0) < f x0f (Vn x2) - f x0 < 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Vn x0H6:forall n : nat, f (Vn n) <= 0Temp:x <= x0 <= yH7:Un_cv (fun i : nat => f (Vn i)) (f x0)Heq:0 = f x0f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Vn x0H6:forall n : nat, f (Vn n) <= 0Temp:x <= x0 <= yH7:Un_cv (fun i : nat => f (Vn i)) (f x0)r:0 > f x0f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RUn_cv Vn x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Vn x0H6:forall n : nat, f (Vn n) <= 0Temp:x <= x0 <= yH7:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> Rabs (f (Vn n) - f x0) < epsHlt:0 < f x0x2:natH8:forall n : nat, (n >= x2)%nat -> Rabs (f (Vn n) - f x0) < f x0H9:(x2 >= x2)%natH10:- (f (Vn x2) - f x0) < f x0 + 0f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Vn x0H6:forall n : nat, f (Vn n) <= 0Temp:x <= x0 <= yH7:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> Rabs (f (Vn n) - f x0) < epsHlt:0 < f x0x2:natH8:forall n : nat, (n >= x2)%nat -> Rabs (f (Vn n) - f x0) < f x0H9:(x2 >= x2)%natH10:Rabs (f (Vn x2) - f x0) < f x0f (Vn x2) - f x0 < 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Vn x0H6:forall n : nat, f (Vn n) <= 0Temp:x <= x0 <= yH7:Un_cv (fun i : nat => f (Vn i)) (f x0)Heq:0 = f x0f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Vn x0H6:forall n : nat, f (Vn n) <= 0Temp:x <= x0 <= yH7:Un_cv (fun i : nat => f (Vn i)) (f x0)r:0 > f x0f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RUn_cv Vn x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Vn x0H6:forall n : nat, f (Vn n) <= 0Temp:x <= x0 <= yH7:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> Rabs (f (Vn n) - f x0) < epsHlt:0 < f x0x2:natH8:forall n : nat, (n >= x2)%nat -> Rabs (f (Vn n) - f x0) < f x0H9:(x2 >= x2)%natH10:f x0 - f (Vn x2) < f x0 + 0f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Vn x0H6:forall n : nat, f (Vn n) <= 0Temp:x <= x0 <= yH7:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> Rabs (f (Vn n) - f x0) < epsHlt:0 < f x0x2:natH8:forall n : nat, (n >= x2)%nat -> Rabs (f (Vn n) - f x0) < f x0H9:(x2 >= x2)%natH10:Rabs (f (Vn x2) - f x0) < f x0f (Vn x2) - f x0 < 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Vn x0H6:forall n : nat, f (Vn n) <= 0Temp:x <= x0 <= yH7:Un_cv (fun i : nat => f (Vn i)) (f x0)Heq:0 = f x0f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Vn x0H6:forall n : nat, f (Vn n) <= 0Temp:x <= x0 <= yH7:Un_cv (fun i : nat => f (Vn i)) (f x0)r:0 > f x0f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RUn_cv Vn x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Vn x0H6:forall n : nat, f (Vn n) <= 0Temp:x <= x0 <= yH7:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> Rabs (f (Vn n) - f x0) < epsHlt:0 < f x0x2:natH8:forall n : nat, (n >= x2)%nat -> Rabs (f (Vn n) - f x0) < f x0H9:(x2 >= x2)%natH10:f x0 + - f (Vn x2) < f x0 + 0f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Vn x0H6:forall n : nat, f (Vn n) <= 0Temp:x <= x0 <= yH7:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> Rabs (f (Vn n) - f x0) < epsHlt:0 < f x0x2:natH8:forall n : nat, (n >= x2)%nat -> Rabs (f (Vn n) - f x0) < f x0H9:(x2 >= x2)%natH10:Rabs (f (Vn x2) - f x0) < f x0f (Vn x2) - f x0 < 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Vn x0H6:forall n : nat, f (Vn n) <= 0Temp:x <= x0 <= yH7:Un_cv (fun i : nat => f (Vn i)) (f x0)Heq:0 = f x0f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Vn x0H6:forall n : nat, f (Vn n) <= 0Temp:x <= x0 <= yH7:Un_cv (fun i : nat => f (Vn i)) (f x0)r:0 > f x0f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RUn_cv Vn x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Vn x0H6:forall n : nat, f (Vn n) <= 0Temp:x <= x0 <= yH7:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> Rabs (f (Vn n) - f x0) < epsHlt:0 < f x0x2:natH8:forall n : nat, (n >= x2)%nat -> Rabs (f (Vn n) - f x0) < f x0H9:(x2 >= x2)%natH10:f x0 + - f (Vn x2) < f x0 + 0H11:- f (Vn x2) < 0f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Vn x0H6:forall n : nat, f (Vn n) <= 0Temp:x <= x0 <= yH7:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> Rabs (f (Vn n) - f x0) < epsHlt:0 < f x0x2:natH8:forall n : nat, (n >= x2)%nat -> Rabs (f (Vn n) - f x0) < f x0H9:(x2 >= x2)%natH10:Rabs (f (Vn x2) - f x0) < f x0f (Vn x2) - f x0 < 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Vn x0H6:forall n : nat, f (Vn n) <= 0Temp:x <= x0 <= yH7:Un_cv (fun i : nat => f (Vn i)) (f x0)Heq:0 = f x0f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Vn x0H6:forall n : nat, f (Vn n) <= 0Temp:x <= x0 <= yH7:Un_cv (fun i : nat => f (Vn i)) (f x0)r:0 > f x0f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RUn_cv Vn x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Vn x0H6:forall n : nat, f (Vn n) <= 0Temp:x <= x0 <= yH7:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> Rabs (f (Vn n) - f x0) < epsHlt:0 < f x0x2:natH8:forall n : nat, (n >= x2)%nat -> Rabs (f (Vn n) - f x0) < f x0H9:(x2 >= x2)%natH10:f x0 + - f (Vn x2) < f x0 + 0H11:- f (Vn x2) < 0H12:f (Vn x2) <= 0f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Vn x0H6:forall n : nat, f (Vn n) <= 0Temp:x <= x0 <= yH7:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> Rabs (f (Vn n) - f x0) < epsHlt:0 < f x0x2:natH8:forall n : nat, (n >= x2)%nat -> Rabs (f (Vn n) - f x0) < f x0H9:(x2 >= x2)%natH10:Rabs (f (Vn x2) - f x0) < f x0f (Vn x2) - f x0 < 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Vn x0H6:forall n : nat, f (Vn n) <= 0Temp:x <= x0 <= yH7:Un_cv (fun i : nat => f (Vn i)) (f x0)Heq:0 = f x0f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Vn x0H6:forall n : nat, f (Vn n) <= 0Temp:x <= x0 <= yH7:Un_cv (fun i : nat => f (Vn i)) (f x0)r:0 > f x0f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RUn_cv Vn x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Vn x0H6:forall n : nat, f (Vn n) <= 0Temp:x <= x0 <= yH7:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> Rabs (f (Vn n) - f x0) < epsHlt:0 < f x0x2:natH8:forall n : nat, (n >= x2)%nat -> Rabs (f (Vn n) - f x0) < f x0H9:(x2 >= x2)%natH10:f x0 + - f (Vn x2) < f x0 + 0H11:- f (Vn x2) < 0H12:f (Vn x2) <= 00 < f (Vn x2) -> f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Vn x0H6:forall n : nat, f (Vn n) <= 0Temp:x <= x0 <= yH7:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> Rabs (f (Vn n) - f x0) < epsHlt:0 < f x0x2:natH8:forall n : nat, (n >= x2)%nat -> Rabs (f (Vn n) - f x0) < f x0H9:(x2 >= x2)%natH10:f x0 + - f (Vn x2) < f x0 + 0H11:- f (Vn x2) < 0H12:f (Vn x2) <= 00 < f (Vn x2)f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Vn x0H6:forall n : nat, f (Vn n) <= 0Temp:x <= x0 <= yH7:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> Rabs (f (Vn n) - f x0) < epsHlt:0 < f x0x2:natH8:forall n : nat, (n >= x2)%nat -> Rabs (f (Vn n) - f x0) < f x0H9:(x2 >= x2)%natH10:Rabs (f (Vn x2) - f x0) < f x0f (Vn x2) - f x0 < 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Vn x0H6:forall n : nat, f (Vn n) <= 0Temp:x <= x0 <= yH7:Un_cv (fun i : nat => f (Vn i)) (f x0)Heq:0 = f x0f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Vn x0H6:forall n : nat, f (Vn n) <= 0Temp:x <= x0 <= yH7:Un_cv (fun i : nat => f (Vn i)) (f x0)r:0 > f x0f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RUn_cv Vn x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Vn x0H6:forall n : nat, f (Vn n) <= 0Temp:x <= x0 <= yH7:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> Rabs (f (Vn n) - f x0) < epsHlt:0 < f x0x2:natH8:forall n : nat, (n >= x2)%nat -> Rabs (f (Vn n) - f x0) < f x0H9:(x2 >= x2)%natH10:f x0 + - f (Vn x2) < f x0 + 0H11:- f (Vn x2) < 0H12:f (Vn x2) <= 0H13:0 < f (Vn x2)f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Vn x0H6:forall n : nat, f (Vn n) <= 0Temp:x <= x0 <= yH7:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> Rabs (f (Vn n) - f x0) < epsHlt:0 < f x0x2:natH8:forall n : nat, (n >= x2)%nat -> Rabs (f (Vn n) - f x0) < f x0H9:(x2 >= x2)%natH10:f x0 + - f (Vn x2) < f x0 + 0H11:- f (Vn x2) < 0H12:f (Vn x2) <= 00 < f (Vn x2)f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Vn x0H6:forall n : nat, f (Vn n) <= 0Temp:x <= x0 <= yH7:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> Rabs (f (Vn n) - f x0) < epsHlt:0 < f x0x2:natH8:forall n : nat, (n >= x2)%nat -> Rabs (f (Vn n) - f x0) < f x0H9:(x2 >= x2)%natH10:Rabs (f (Vn x2) - f x0) < f x0f (Vn x2) - f x0 < 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Vn x0H6:forall n : nat, f (Vn n) <= 0Temp:x <= x0 <= yH7:Un_cv (fun i : nat => f (Vn i)) (f x0)Heq:0 = f x0f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Vn x0H6:forall n : nat, f (Vn n) <= 0Temp:x <= x0 <= yH7:Un_cv (fun i : nat => f (Vn i)) (f x0)r:0 > f x0f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RUn_cv Vn x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Vn x0H6:forall n : nat, f (Vn n) <= 0Temp:x <= x0 <= yH7:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> Rabs (f (Vn n) - f x0) < epsHlt:0 < f x0x2:natH8:forall n : nat, (n >= x2)%nat -> Rabs (f (Vn n) - f x0) < f x0H9:(x2 >= x2)%natH10:f x0 + - f (Vn x2) < f x0 + 0H11:- f (Vn x2) < 0H12:f (Vn x2) <= 00 < f (Vn x2)f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Vn x0H6:forall n : nat, f (Vn n) <= 0Temp:x <= x0 <= yH7:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> Rabs (f (Vn n) - f x0) < epsHlt:0 < f x0x2:natH8:forall n : nat, (n >= x2)%nat -> Rabs (f (Vn n) - f x0) < f x0H9:(x2 >= x2)%natH10:Rabs (f (Vn x2) - f x0) < f x0f (Vn x2) - f x0 < 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Vn x0H6:forall n : nat, f (Vn n) <= 0Temp:x <= x0 <= yH7:Un_cv (fun i : nat => f (Vn i)) (f x0)Heq:0 = f x0f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Vn x0H6:forall n : nat, f (Vn n) <= 0Temp:x <= x0 <= yH7:Un_cv (fun i : nat => f (Vn i)) (f x0)r:0 > f x0f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RUn_cv Vn x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Vn x0H6:forall n : nat, f (Vn n) <= 0Temp:x <= x0 <= yH7:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> Rabs (f (Vn n) - f x0) < epsHlt:0 < f x0x2:natH8:forall n : nat, (n >= x2)%nat -> Rabs (f (Vn n) - f x0) < f x0H9:(x2 >= x2)%natH10:f x0 + - f (Vn x2) < f x0 + 0H11:- f (Vn x2) < 0H12:f (Vn x2) <= 00 < - - f (Vn x2)f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Vn x0H6:forall n : nat, f (Vn n) <= 0Temp:x <= x0 <= yH7:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> Rabs (f (Vn n) - f x0) < epsHlt:0 < f x0x2:natH8:forall n : nat, (n >= x2)%nat -> Rabs (f (Vn n) - f x0) < f x0H9:(x2 >= x2)%natH10:Rabs (f (Vn x2) - f x0) < f x0f (Vn x2) - f x0 < 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Vn x0H6:forall n : nat, f (Vn n) <= 0Temp:x <= x0 <= yH7:Un_cv (fun i : nat => f (Vn i)) (f x0)Heq:0 = f x0f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Vn x0H6:forall n : nat, f (Vn n) <= 0Temp:x <= x0 <= yH7:Un_cv (fun i : nat => f (Vn i)) (f x0)r:0 > f x0f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RUn_cv Vn x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Vn x0H6:forall n : nat, f (Vn n) <= 0Temp:x <= x0 <= yH7:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> Rabs (f (Vn n) - f x0) < epsHlt:0 < f x0x2:natH8:forall n : nat, (n >= x2)%nat -> Rabs (f (Vn n) - f x0) < f x0H9:(x2 >= x2)%natH10:Rabs (f (Vn x2) - f x0) < f x0f (Vn x2) - f x0 < 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Vn x0H6:forall n : nat, f (Vn n) <= 0Temp:x <= x0 <= yH7:Un_cv (fun i : nat => f (Vn i)) (f x0)Heq:0 = f x0f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Vn x0H6:forall n : nat, f (Vn n) <= 0Temp:x <= x0 <= yH7:Un_cv (fun i : nat => f (Vn i)) (f x0)r:0 > f x0f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RUn_cv Vn x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Vn x0H6:forall n : nat, f (Vn n) <= 0Temp:x <= x0 <= yH7:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> Rabs (f (Vn n) - f x0) < epsHlt:0 < f x0x2:natH8:forall n : nat, (n >= x2)%nat -> Rabs (f (Vn n) - f x0) < f x0H9:(x2 >= x2)%natH10:Rabs (f (Vn x2) - f x0) < f x0f x0 - f (Vn x2) + (f (Vn x2) - f x0) < f x0 - f (Vn x2) + 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Vn x0H6:forall n : nat, f (Vn n) <= 0Temp:x <= x0 <= yH7:Un_cv (fun i : nat => f (Vn i)) (f x0)Heq:0 = f x0f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Vn x0H6:forall n : nat, f (Vn n) <= 0Temp:x <= x0 <= yH7:Un_cv (fun i : nat => f (Vn i)) (f x0)r:0 > f x0f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RUn_cv Vn x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Vn x0H6:forall n : nat, f (Vn n) <= 0Temp:x <= x0 <= yH7:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> Rabs (f (Vn n) - f x0) < epsHlt:0 < f x0x2:natH8:forall n : nat, (n >= x2)%nat -> Rabs (f (Vn n) - f x0) < f x0H9:(x2 >= x2)%natH10:Rabs (f (Vn x2) - f x0) < f x00 < f x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Vn x0H6:forall n : nat, f (Vn n) <= 0Temp:x <= x0 <= yH7:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> Rabs (f (Vn n) - f x0) < epsHlt:0 < f x0x2:natH8:forall n : nat, (n >= x2)%nat -> Rabs (f (Vn n) - f x0) < f x0H9:(x2 >= x2)%natH10:Rabs (f (Vn x2) - f x0) < f x00 <= - f (Vn x2)f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Vn x0H6:forall n : nat, f (Vn n) <= 0Temp:x <= x0 <= yH7:Un_cv (fun i : nat => f (Vn i)) (f x0)Heq:0 = f x0f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Vn x0H6:forall n : nat, f (Vn n) <= 0Temp:x <= x0 <= yH7:Un_cv (fun i : nat => f (Vn i)) (f x0)r:0 > f x0f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RUn_cv Vn x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Vn x0H6:forall n : nat, f (Vn n) <= 0Temp:x <= x0 <= yH7:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> Rabs (f (Vn n) - f x0) < epsHlt:0 < f x0x2:natH8:forall n : nat, (n >= x2)%nat -> Rabs (f (Vn n) - f x0) < f x0H9:(x2 >= x2)%natH10:Rabs (f (Vn x2) - f x0) < f x00 <= - f (Vn x2)f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Vn x0H6:forall n : nat, f (Vn n) <= 0Temp:x <= x0 <= yH7:Un_cv (fun i : nat => f (Vn i)) (f x0)Heq:0 = f x0f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Vn x0H6:forall n : nat, f (Vn n) <= 0Temp:x <= x0 <= yH7:Un_cv (fun i : nat => f (Vn i)) (f x0)r:0 > f x0f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RUn_cv Vn x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Vn x0H6:forall n : nat, f (Vn n) <= 0Temp:x <= x0 <= yH7:Un_cv (fun i : nat => f (Vn i)) (f x0)Heq:0 = f x0f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Vn x0H6:forall n : nat, f (Vn n) <= 0Temp:x <= x0 <= yH7:Un_cv (fun i : nat => f (Vn i)) (f x0)r:0 > f x0f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RUn_cv Vn x0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RH5:Un_cv Vn x0H6:forall n : nat, f (Vn n) <= 0Temp:x <= x0 <= yH7:Un_cv (fun i : nat => f (Vn i)) (f x0)r:0 > f x0f x0 <= 0f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RUn_cv Vn x0unfold Vn in |- *; assumption. Qed. (* begin hide *) Ltac case_le H := let t := type of H in let h' := fresh in match t with ?x <= ?y => case (total_order_T x y); [intros h'; case h'; clear h' | intros h'; clear -H h'; elimtype False; lra ] end. (* end hide *)f:R -> Rx, y:RH:forall a : R, x <= a <= y -> continuity_pt f aH0:x < yH1:f x < 0H2:0 < f yH3:x <= yX:{l : R | Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) l}X0:{l : R | Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) l}x0:Rp:Un_cv (dicho_up x y (fun z : R => cond_positivity (f z))) x0x1:Rp0:Un_cv (dicho_lb x y (fun z : R => cond_positivity (f z))) x0H4:x1 = x0Vn:=fun n : nat => dicho_lb x y (fun z : R => cond_positivity (f z)) n:nat -> RWn:=fun n : nat => dicho_up x y (fun z : R => cond_positivity (f z)) n:nat -> RUn_cv Vn x0forall (f : R -> R) (lb ub y : R), lb < ub -> f lb <= y <= f ub -> (forall x : R, lb <= x <= ub -> continuity_pt f x) -> {x : R | lb <= x <= ub /\ f x = y}forall (f : R -> R) (lb ub y : R), lb < ub -> f lb <= y <= f ub -> (forall x : R, lb <= x <= ub -> continuity_pt f x) -> {x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f x{x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy <= f ub -> {x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb < yy_encad3:y <= f uby < f ub -> {x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb < yy_encad3:y <= f uby = f ub -> {x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb = yy_encad3:y <= f uby < f ub -> {x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb = yy_encad3:y <= f uby = f ub -> {x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb < yy_encad3:y <= f uby_encad4:y < f ub{x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb < yy_encad3:y <= f uby = f ub -> {x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb = yy_encad3:y <= f uby < f ub -> {x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb = yy_encad3:y <= f uby = f ub -> {x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad2:f lb < yy_encad4:y < f ub{x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb < yy_encad3:y <= f uby = f ub -> {x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb = yy_encad3:y <= f uby < f ub -> {x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb = yy_encad3:y <= f uby = f ub -> {x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad2:f lb < yy_encad4:y < f ubforall a : R, lb <= a <= ub -> continuity_pt (fun x : R => f x - y) af:R -> Rlb, ub, y:Rlb_lt_ub:lb < ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad2:f lb < yy_encad4:y < f ubCont:forall a : R, lb <= a <= ub -> continuity_pt (fun x : R => f x - y) a{x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb < yy_encad3:y <= f uby = f ub -> {x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb = yy_encad3:y <= f uby < f ub -> {x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb = yy_encad3:y <= f uby = f ub -> {x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad2:f lb < yy_encad4:y < f uba:Ra_encad:lb <= a <= ubcontinuity_pt (fun x : R => f x - y) af:R -> Rlb, ub, y:Rlb_lt_ub:lb < ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad2:f lb < yy_encad4:y < f ubCont:forall a : R, lb <= a <= ub -> continuity_pt (fun x : R => f x - y) a{x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb < yy_encad3:y <= f uby = f ub -> {x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb = yy_encad3:y <= f uby < f ub -> {x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb = yy_encad3:y <= f uby = f ub -> {x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad2:f lb < yy_encad4:y < f uba:Ra_encad:lb <= a <= ubforall eps : R, eps > 0 -> exists alp : R, alp > 0 /\ (forall x : R, D_x no_cond a x /\ Rabs (x - a) < alp -> Rabs (f x - y - (f a - y)) < eps)f:R -> Rlb, ub, y:Rlb_lt_ub:lb < ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad2:f lb < yy_encad4:y < f ubCont:forall a : R, lb <= a <= ub -> continuity_pt (fun x : R => f x - y) a{x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb < yy_encad3:y <= f uby = f ub -> {x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb = yy_encad3:y <= f uby < f ub -> {x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb = yy_encad3:y <= f uby = f ub -> {x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad2:f lb < yy_encad4:y < f uba:Ra_encad:lb <= a <= ubeps:Reps_pos:eps > 0exists alp : R, alp > 0 /\ (forall x : R, D_x no_cond a x /\ Rabs (x - a) < alp -> Rabs (f x - y - (f a - y)) < eps)f:R -> Rlb, ub, y:Rlb_lt_ub:lb < ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad2:f lb < yy_encad4:y < f ubCont:forall a : R, lb <= a <= ub -> continuity_pt (fun x : R => f x - y) a{x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb < yy_encad3:y <= f uby = f ub -> {x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb = yy_encad3:y <= f uby < f ub -> {x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb = yy_encad3:y <= f uby = f ub -> {x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad2:f lb < yy_encad4:y < f uba:Ra_encad:lb <= a <= ubeps:Reps_pos:eps > 0forall x : R, x > 0 /\ (forall x0 : Base R_met, D_x no_cond a x0 /\ dist R_met x0 a < x -> dist R_met (f x0) (f a) < eps) -> exists alp : R, alp > 0 /\ (forall x0 : R, D_x no_cond a x0 /\ Rabs (x0 - a) < alp -> Rabs (f x0 - y - (f a - y)) < eps)f:R -> Rlb, ub, y:Rlb_lt_ub:lb < ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad2:f lb < yy_encad4:y < f ubCont:forall a : R, lb <= a <= ub -> continuity_pt (fun x : R => f x - y) a{x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb < yy_encad3:y <= f uby = f ub -> {x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb = yy_encad3:y <= f uby < f ub -> {x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb = yy_encad3:y <= f uby = f ub -> {x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad2:f lb < yy_encad4:y < f uba:Ra_encad:lb <= a <= ubeps:Reps_pos:eps > 0alpha:Ralpha_pos:alpha > 0 /\ (forall x : Base R_met, D_x no_cond a x /\ dist R_met x a < alpha -> dist R_met (f x) (f a) < eps)exists alp : R, alp > 0 /\ (forall x : R, D_x no_cond a x /\ Rabs (x - a) < alp -> Rabs (f x - y - (f a - y)) < eps)f:R -> Rlb, ub, y:Rlb_lt_ub:lb < ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad2:f lb < yy_encad4:y < f ubCont:forall a : R, lb <= a <= ub -> continuity_pt (fun x : R => f x - y) a{x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb < yy_encad3:y <= f uby = f ub -> {x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb = yy_encad3:y <= f uby < f ub -> {x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb = yy_encad3:y <= f uby = f ub -> {x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad2:f lb < yy_encad4:y < f uba:Ra_encad:lb <= a <= ubeps:Reps_pos:eps > 0alpha:Ralpha_pos:alpha > 0Temp:forall x : Base R_met, D_x no_cond a x /\ dist R_met x a < alpha -> dist R_met (f x) (f a) < epsexists alp : R, alp > 0 /\ (forall x : R, D_x no_cond a x /\ Rabs (x - a) < alp -> Rabs (f x - y - (f a - y)) < eps)f:R -> Rlb, ub, y:Rlb_lt_ub:lb < ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad2:f lb < yy_encad4:y < f ubCont:forall a : R, lb <= a <= ub -> continuity_pt (fun x : R => f x - y) a{x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb < yy_encad3:y <= f uby = f ub -> {x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb = yy_encad3:y <= f uby < f ub -> {x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb = yy_encad3:y <= f uby = f ub -> {x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad2:f lb < yy_encad4:y < f uba:Ra_encad:lb <= a <= ubeps:Reps_pos:eps > 0alpha:Ralpha_pos:alpha > 0Temp:forall x : Base R_met, D_x no_cond a x /\ dist R_met x a < alpha -> dist R_met (f x) (f a) < epsalpha > 0 /\ (forall x : R, D_x no_cond a x /\ Rabs (x - a) < alpha -> Rabs (f x - y - (f a - y)) < eps)f:R -> Rlb, ub, y:Rlb_lt_ub:lb < ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad2:f lb < yy_encad4:y < f ubCont:forall a : R, lb <= a <= ub -> continuity_pt (fun x : R => f x - y) a{x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb < yy_encad3:y <= f uby = f ub -> {x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb = yy_encad3:y <= f uby < f ub -> {x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb = yy_encad3:y <= f uby = f ub -> {x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad2:f lb < yy_encad4:y < f uba:Ra_encad:lb <= a <= ubeps:Reps_pos:eps > 0alpha:Ralpha_pos:alpha > 0Temp:forall x : Base R_met, D_x no_cond a x /\ dist R_met x a < alpha -> dist R_met (f x) (f a) < epsalpha > 0f:R -> Rlb, ub, y:Rlb_lt_ub:lb < ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad2:f lb < yy_encad4:y < f uba:Ra_encad:lb <= a <= ubeps:Reps_pos:eps > 0alpha:Ralpha_pos:alpha > 0Temp:forall x : Base R_met, D_x no_cond a x /\ dist R_met x a < alpha -> dist R_met (f x) (f a) < epsforall x : R, D_x no_cond a x /\ Rabs (x - a) < alpha -> Rabs (f x - y - (f a - y)) < epsf:R -> Rlb, ub, y:Rlb_lt_ub:lb < ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad2:f lb < yy_encad4:y < f ubCont:forall a : R, lb <= a <= ub -> continuity_pt (fun x : R => f x - y) a{x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb < yy_encad3:y <= f uby = f ub -> {x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb = yy_encad3:y <= f uby < f ub -> {x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb = yy_encad3:y <= f uby = f ub -> {x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad2:f lb < yy_encad4:y < f uba:Ra_encad:lb <= a <= ubeps:Reps_pos:eps > 0alpha:Ralpha_pos:alpha > 0Temp:forall x : Base R_met, D_x no_cond a x /\ dist R_met x a < alpha -> dist R_met (f x) (f a) < epsforall x : R, D_x no_cond a x /\ Rabs (x - a) < alpha -> Rabs (f x - y - (f a - y)) < epsf:R -> Rlb, ub, y:Rlb_lt_ub:lb < ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad2:f lb < yy_encad4:y < f ubCont:forall a : R, lb <= a <= ub -> continuity_pt (fun x : R => f x - y) a{x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb < yy_encad3:y <= f uby = f ub -> {x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb = yy_encad3:y <= f uby < f ub -> {x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb = yy_encad3:y <= f uby = f ub -> {x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < ubf_cont_interv:forall x0 : R, lb <= x0 <= ub -> continuity_pt f x0y_encad2:f lb < yy_encad4:y < f uba:Ra_encad:lb <= a <= ubeps:Reps_pos:eps > 0alpha:Ralpha_pos:alpha > 0Temp:forall x0 : Base R_met, D_x no_cond a x0 /\ dist R_met x0 a < alpha -> dist R_met (f x0) (f a) < epsx:Rx_cond:D_x no_cond a x /\ Rabs (x - a) < alphaRabs (f x - y - (f a - y)) < epsf:R -> Rlb, ub, y:Rlb_lt_ub:lb < ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad2:f lb < yy_encad4:y < f ubCont:forall a : R, lb <= a <= ub -> continuity_pt (fun x : R => f x - y) a{x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb < yy_encad3:y <= f uby = f ub -> {x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb = yy_encad3:y <= f uby < f ub -> {x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb = yy_encad3:y <= f uby = f ub -> {x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < ubf_cont_interv:forall x0 : R, lb <= x0 <= ub -> continuity_pt f x0y_encad2:f lb < yy_encad4:y < f uba:Ra_encad:lb <= a <= ubeps:Reps_pos:eps > 0alpha:Ralpha_pos:alpha > 0Temp:forall x0 : Base R_met, D_x no_cond a x0 /\ dist R_met x0 a < alpha -> dist R_met (f x0) (f a) < epsx:Rx_cond:D_x no_cond a x /\ Rabs (x - a) < alphaRabs (f x - f a) < epsf:R -> Rlb, ub, y:Rlb_lt_ub:lb < ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad2:f lb < yy_encad4:y < f ubCont:forall a : R, lb <= a <= ub -> continuity_pt (fun x : R => f x - y) a{x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb < yy_encad3:y <= f uby = f ub -> {x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb = yy_encad3:y <= f uby < f ub -> {x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb = yy_encad3:y <= f uby = f ub -> {x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad2:f lb < yy_encad4:y < f ubCont:forall a : R, lb <= a <= ub -> continuity_pt (fun x : R => f x - y) a{x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb < yy_encad3:y <= f uby = f ub -> {x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb = yy_encad3:y <= f uby < f ub -> {x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb = yy_encad3:y <= f uby = f ub -> {x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad2:f lb < yy_encad4:y < f ubCont:forall a : R, lb <= a <= ub -> continuity_pt (fun x : R => f x - y) af lb - y < 0f:R -> Rlb, ub, y:Rlb_lt_ub:lb < ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad2:f lb < yy_encad4:y < f ubCont:forall a : R, lb <= a <= ub -> continuity_pt (fun x : R => f x - y) aH1:(fun x : R => f x - y) lb < 0{x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb < yy_encad3:y <= f uby = f ub -> {x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb = yy_encad3:y <= f uby < f ub -> {x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb = yy_encad3:y <= f uby = f ub -> {x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad2:f lb < yy_encad4:y < f ubCont:forall a : R, lb <= a <= ub -> continuity_pt (fun x : R => f x - y) af lb < yf:R -> Rlb, ub, y:Rlb_lt_ub:lb < ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad2:f lb < yy_encad4:y < f ubCont:forall a : R, lb <= a <= ub -> continuity_pt (fun x : R => f x - y) aH1:(fun x : R => f x - y) lb < 0{x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb < yy_encad3:y <= f uby = f ub -> {x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb = yy_encad3:y <= f uby < f ub -> {x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb = yy_encad3:y <= f uby = f ub -> {x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad2:f lb < yy_encad4:y < f ubCont:forall a : R, lb <= a <= ub -> continuity_pt (fun x : R => f x - y) aH1:(fun x : R => f x - y) lb < 0{x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb < yy_encad3:y <= f uby = f ub -> {x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb = yy_encad3:y <= f uby < f ub -> {x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb = yy_encad3:y <= f uby = f ub -> {x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad2:f lb < yy_encad4:y < f ubCont:forall a : R, lb <= a <= ub -> continuity_pt (fun x : R => f x - y) aH1:(fun x : R => f x - y) lb < 00 < f ub - yf:R -> Rlb, ub, y:Rlb_lt_ub:lb < ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad2:f lb < yy_encad4:y < f ubCont:forall a : R, lb <= a <= ub -> continuity_pt (fun x : R => f x - y) aH1:(fun x : R => f x - y) lb < 0H2:0 < (fun x : R => f x - y) ub{x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb < yy_encad3:y <= f uby = f ub -> {x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb = yy_encad3:y <= f uby < f ub -> {x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb = yy_encad3:y <= f uby = f ub -> {x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad2:f lb < yy_encad4:y < f ubCont:forall a : R, lb <= a <= ub -> continuity_pt (fun x : R => f x - y) aH1:(fun x : R => f x - y) lb < 0H2:0 < (fun x : R => f x - y) ub{x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb < yy_encad3:y <= f uby = f ub -> {x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb = yy_encad3:y <= f uby < f ub -> {x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb = yy_encad3:y <= f uby = f ub -> {x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < ubf_cont_interv:forall x0 : R, lb <= x0 <= ub -> continuity_pt f x0y_encad2:f lb < yy_encad4:y < f ubCont:forall a : R, lb <= a <= ub -> continuity_pt (fun x0 : R => f x0 - y) aH1:(fun x0 : R => f x0 - y) lb < 0H2:0 < (fun x0 : R => f x0 - y) ubx:RHx:lb <= x <= ub /\ f x - y = 0{x0 : R | lb <= x0 <= ub /\ f x0 = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb < yy_encad3:y <= f uby = f ub -> {x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb = yy_encad3:y <= f uby < f ub -> {x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb = yy_encad3:y <= f uby = f ub -> {x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < ubf_cont_interv:forall x0 : R, lb <= x0 <= ub -> continuity_pt f x0y_encad2:f lb < yy_encad4:y < f ubCont:forall a : R, lb <= a <= ub -> continuity_pt (fun x0 : R => f x0 - y) aH1:(fun x0 : R => f x0 - y) lb < 0H2:0 < (fun x0 : R => f x0 - y) ubx:RHx:lb <= x <= ub /\ f x - y = 0lb <= x <= ub /\ f x = yf:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb < yy_encad3:y <= f uby = f ub -> {x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb = yy_encad3:y <= f uby < f ub -> {x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb = yy_encad3:y <= f uby = f ub -> {x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < ubf_cont_interv:forall x0 : R, lb <= x0 <= ub -> continuity_pt f x0y_encad2:f lb < yy_encad4:y < f ubCont:forall a : R, lb <= a <= ub -> continuity_pt (fun x0 : R => f x0 - y) aH1:(fun x0 : R => f x0 - y) lb < 0H2:0 < (fun x0 : R => f x0 - y) ubx:RHyp:lb <= x <= ubResult:f x - y = 0lb <= x <= ub /\ f x = yf:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb < yy_encad3:y <= f uby = f ub -> {x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb = yy_encad3:y <= f uby < f ub -> {x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb = yy_encad3:y <= f uby = f ub -> {x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb < yy_encad3:y <= f uby = f ub -> {x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb = yy_encad3:y <= f uby < f ub -> {x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb = yy_encad3:y <= f uby = f ub -> {x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb = yy_encad3:y <= f uby < f ub -> {x : R | lb <= x <= ub /\ f x = y}f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb = yy_encad3:y <= f uby = f ub -> {x : R | lb <= x <= ub /\ f x = y}intro H ; exists ub ; intuition. Qed.f:R -> Rlb, ub, y:Rlb_lt_ub:lb < uby_encad:f lb <= y <= f ubf_cont_interv:forall x : R, lb <= x <= ub -> continuity_pt f xy_encad1:f lb <= yy_encad2:f lb = yy_encad3:y <= f uby = f ub -> {x : R | lb <= x <= ub /\ f x = y}
forall (f g : R -> R) (lb ub : R), lb < ub -> (forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f y) -> (forall x : R, lb <= x <= ub -> comp g f x = id x) -> (forall a : R, lb <= a <= ub -> continuity_pt f a) -> forall b : R, f lb < b < f ub -> continuity_pt g bforall (f g : R -> R) (lb ub : R), lb < ub -> (forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f y) -> (forall x : R, lb <= x <= ub -> comp g f x = id x) -> (forall a : R, lb <= a <= ub -> continuity_pt f a) -> forall b : R, f lb < b < f ub -> continuity_pt g bforall x y z : R, Rmax x y < z <-> x < z /\ y < zSublemma:forall x y z : R, Rmax x y < z <-> x < z /\ y < zforall (f g : R -> R) (lb ub : R), lb < ub -> (forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f y) -> (forall x : R, lb <= x <= ub -> comp g f x = id x) -> (forall a : R, lb <= a <= ub -> continuity_pt f a) -> forall b : R, f lb < b < f ub -> continuity_pt g bx, y, z:RRmax x y < z <-> x < z /\ y < zSublemma:forall x y z : R, Rmax x y < z <-> x < z /\ y < zforall (f g : R -> R) (lb ub : R), lb < ub -> (forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f y) -> (forall x : R, lb <= x <= ub -> comp g f x = id x) -> (forall a : R, lb <= a <= ub -> continuity_pt f a) -> forall b : R, f lb < b < f ub -> continuity_pt g bx, y, z:RRmax x y < z -> x < z /\ y < zx, y, z:Rx < z /\ y < z -> Rmax x y < zSublemma:forall x y z : R, Rmax x y < z <-> x < z /\ y < zforall (f g : R -> R) (lb ub : R), lb < ub -> (forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f y) -> (forall x : R, lb <= x <= ub -> comp g f x = id x) -> (forall a : R, lb <= a <= ub -> continuity_pt f a) -> forall b : R, f lb < b < f ub -> continuity_pt g bx, y, z:R(if Rle_dec x y then y else x) < z -> x < z /\ y < zx, y, z:Rx < z /\ y < z -> Rmax x y < zSublemma:forall x y z : R, Rmax x y < z <-> x < z /\ y < zforall (f g : R -> R) (lb ub : R), lb < ub -> (forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f y) -> (forall x : R, lb <= x <= ub -> comp g f x = id x) -> (forall a : R, lb <= a <= ub -> continuity_pt f a) -> forall b : R, f lb < b < f ub -> continuity_pt g bx, y, z:RHyp:x <= yHyp2:y < zx < z /\ y < zx, y, z:RHyp:~ x <= yHyp2:x < zx < z /\ y < zx, y, z:Rx < z /\ y < z -> Rmax x y < zSublemma:forall x y z : R, Rmax x y < z <-> x < z /\ y < zforall (f g : R -> R) (lb ub : R), lb < ub -> (forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f y) -> (forall x : R, lb <= x <= ub -> comp g f x = id x) -> (forall a : R, lb <= a <= ub -> continuity_pt f a) -> forall b : R, f lb < b < f ub -> continuity_pt g bx, y, z:RHyp:x <= yHyp2:y < zx < zx, y, z:RHyp:x <= yHyp2:y < zy < zx, y, z:RHyp:~ x <= yHyp2:x < zx < z /\ y < zx, y, z:Rx < z /\ y < z -> Rmax x y < zSublemma:forall x y z : R, Rmax x y < z <-> x < z /\ y < zforall (f g : R -> R) (lb ub : R), lb < ub -> (forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f y) -> (forall x : R, lb <= x <= ub -> comp g f x = id x) -> (forall a : R, lb <= a <= ub -> continuity_pt f a) -> forall b : R, f lb < b < f ub -> continuity_pt g bx, y, z:RHyp:x <= yHyp2:y < zy < zx, y, z:RHyp:~ x <= yHyp2:x < zx < z /\ y < zx, y, z:Rx < z /\ y < z -> Rmax x y < zSublemma:forall x y z : R, Rmax x y < z <-> x < z /\ y < zforall (f g : R -> R) (lb ub : R), lb < ub -> (forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f y) -> (forall x : R, lb <= x <= ub -> comp g f x = id x) -> (forall a : R, lb <= a <= ub -> continuity_pt f a) -> forall b : R, f lb < b < f ub -> continuity_pt g bx, y, z:RHyp:~ x <= yHyp2:x < zx < z /\ y < zx, y, z:Rx < z /\ y < z -> Rmax x y < zSublemma:forall x y z : R, Rmax x y < z <-> x < z /\ y < zforall (f g : R -> R) (lb ub : R), lb < ub -> (forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f y) -> (forall x : R, lb <= x <= ub -> comp g f x = id x) -> (forall a : R, lb <= a <= ub -> continuity_pt f a) -> forall b : R, f lb < b < f ub -> continuity_pt g bx, y, z:RHyp:~ x <= yHyp2:x < zx < zx, y, z:RHyp:~ x <= yHyp2:x < zy < zx, y, z:Rx < z /\ y < z -> Rmax x y < zSublemma:forall x y z : R, Rmax x y < z <-> x < z /\ y < zforall (f g : R -> R) (lb ub : R), lb < ub -> (forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f y) -> (forall x : R, lb <= x <= ub -> comp g f x = id x) -> (forall a : R, lb <= a <= ub -> continuity_pt f a) -> forall b : R, f lb < b < f ub -> continuity_pt g bx, y, z:RHyp:~ x <= yHyp2:x < zy < zx, y, z:Rx < z /\ y < z -> Rmax x y < zSublemma:forall x y z : R, Rmax x y < z <-> x < z /\ y < zforall (f g : R -> R) (lb ub : R), lb < ub -> (forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f y) -> (forall x : R, lb <= x <= ub -> comp g f x = id x) -> (forall a : R, lb <= a <= ub -> continuity_pt f a) -> forall b : R, f lb < b < f ub -> continuity_pt g bx, y, z:RHyp:~ x <= yHyp2:x < zy < xx, y, z:RHyp:~ x <= yHyp2:x < zx < zx, y, z:Rx < z /\ y < z -> Rmax x y < zSublemma:forall x y z : R, Rmax x y < z <-> x < z /\ y < zforall (f g : R -> R) (lb ub : R), lb < ub -> (forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f y) -> (forall x : R, lb <= x <= ub -> comp g f x = id x) -> (forall a : R, lb <= a <= ub -> continuity_pt f a) -> forall b : R, f lb < b < f ub -> continuity_pt g bx, y, z:RHyp:~ x <= yHyp2:x < zforall x0 y0 : R, ~ x0 <= y0 -> x0 > y0x, y, z:RHyp:~ x <= yHyp2:x < zTemp:forall x0 y0 : R, ~ x0 <= y0 -> x0 > y0y < xx, y, z:RHyp:~ x <= yHyp2:x < zx < zx, y, z:Rx < z /\ y < z -> Rmax x y < zSublemma:forall x y z : R, Rmax x y < z <-> x < z /\ y < zforall (f g : R -> R) (lb ub : R), lb < ub -> (forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f y) -> (forall x : R, lb <= x <= ub -> comp g f x = id x) -> (forall a : R, lb <= a <= ub -> continuity_pt f a) -> forall b : R, f lb < b < f ub -> continuity_pt g bx, y, z:RHyp:~ x <= yHyp2:x < zm, n:RHypmn:~ m <= nm > nx, y, z:RHyp:~ x <= yHyp2:x < zTemp:forall x0 y0 : R, ~ x0 <= y0 -> x0 > y0y < xx, y, z:RHyp:~ x <= yHyp2:x < zx < zx, y, z:Rx < z /\ y < z -> Rmax x y < zSublemma:forall x y z : R, Rmax x y < z <-> x < z /\ y < zforall (f g : R -> R) (lb ub : R), lb < ub -> (forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f y) -> (forall x : R, lb <= x <= ub -> comp g f x = id x) -> (forall a : R, lb <= a <= ub -> continuity_pt f a) -> forall b : R, f lb < b < f ub -> continuity_pt g bx, y, z:RHyp:~ x <= yHyp2:x < zTemp:forall x0 y0 : R, ~ x0 <= y0 -> x0 > y0y < xx, y, z:RHyp:~ x <= yHyp2:x < zx < zx, y, z:Rx < z /\ y < z -> Rmax x y < zSublemma:forall x y z : R, Rmax x y < z <-> x < z /\ y < zforall (f g : R -> R) (lb ub : R), lb < ub -> (forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f y) -> (forall x : R, lb <= x <= ub -> comp g f x = id x) -> (forall a : R, lb <= a <= ub -> continuity_pt f a) -> forall b : R, f lb < b < f ub -> continuity_pt g bx, y, z:RHyp:~ x <= yHyp2:x < zx < zx, y, z:Rx < z /\ y < z -> Rmax x y < zSublemma:forall x y z : R, Rmax x y < z <-> x < z /\ y < zforall (f g : R -> R) (lb ub : R), lb < ub -> (forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f y) -> (forall x : R, lb <= x <= ub -> comp g f x = id x) -> (forall a : R, lb <= a <= ub -> continuity_pt f a) -> forall b : R, f lb < b < f ub -> continuity_pt g bx, y, z:Rx < z /\ y < z -> Rmax x y < zSublemma:forall x y z : R, Rmax x y < z <-> x < z /\ y < zforall (f g : R -> R) (lb ub : R), lb < ub -> (forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f y) -> (forall x : R, lb <= x <= ub -> comp g f x = id x) -> (forall a : R, lb <= a <= ub -> continuity_pt f a) -> forall b : R, f lb < b < f ub -> continuity_pt g bx, y, z:RHyp:x < z /\ y < zRmax x y < zSublemma:forall x y z : R, Rmax x y < z <-> x < z /\ y < zforall (f g : R -> R) (lb ub : R), lb < ub -> (forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f y) -> (forall x : R, lb <= x <= ub -> comp g f x = id x) -> (forall a : R, lb <= a <= ub -> continuity_pt f a) -> forall b : R, f lb < b < f ub -> continuity_pt g bx, y, z:RHyp:x < z /\ y < z(if Rle_dec x y then y else x) < zSublemma:forall x y z : R, Rmax x y < z <-> x < z /\ y < zforall (f g : R -> R) (lb ub : R), lb < ub -> (forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f y) -> (forall x : R, lb <= x <= ub -> comp g f x = id x) -> (forall a : R, lb <= a <= ub -> continuity_pt f a) -> forall b : R, f lb < b < f ub -> continuity_pt g bx, y, z:RHyp:x < z /\ y < zx <= y -> y < zx, y, z:RHyp:x < z /\ y < z~ x <= y -> x < zSublemma:forall x y z : R, Rmax x y < z <-> x < z /\ y < zforall (f g : R -> R) (lb ub : R), lb < ub -> (forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f y) -> (forall x : R, lb <= x <= ub -> comp g f x = id x) -> (forall a : R, lb <= a <= ub -> continuity_pt f a) -> forall b : R, f lb < b < f ub -> continuity_pt g bx, y, z:RHyp:x < z /\ y < z~ x <= y -> x < zSublemma:forall x y z : R, Rmax x y < z <-> x < z /\ y < zforall (f g : R -> R) (lb ub : R), lb < ub -> (forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f y) -> (forall x : R, lb <= x <= ub -> comp g f x = id x) -> (forall a : R, lb <= a <= ub -> continuity_pt f a) -> forall b : R, f lb < b < f ub -> continuity_pt g bSublemma:forall x y z : R, Rmax x y < z <-> x < z /\ y < zforall (f g : R -> R) (lb ub : R), lb < ub -> (forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f y) -> (forall x : R, lb <= x <= ub -> comp g f x = id x) -> (forall a : R, lb <= a <= ub -> continuity_pt f a) -> forall b : R, f lb < b < f ub -> continuity_pt g bSublemma:forall x y z : R, Rmax x y < z <-> x < z /\ y < zforall x y z : R, Rmin x y > z <-> x > z /\ y > zSublemma:forall x y z : R, Rmax x y < z <-> x < z /\ y < zSublemma2:forall x y z : R, Rmin x y > z <-> x > z /\ y > zforall (f g : R -> R) (lb ub : R), lb < ub -> (forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f y) -> (forall x : R, lb <= x <= ub -> comp g f x = id x) -> (forall a : R, lb <= a <= ub -> continuity_pt f a) -> forall b : R, f lb < b < f ub -> continuity_pt g bSublemma:forall x0 y0 z0 : R, Rmax x0 y0 < z0 <-> x0 < z0 /\ y0 < z0x, y, z:RRmin x y > z <-> x > z /\ y > zSublemma:forall x y z : R, Rmax x y < z <-> x < z /\ y < zSublemma2:forall x y z : R, Rmin x y > z <-> x > z /\ y > zforall (f g : R -> R) (lb ub : R), lb < ub -> (forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f y) -> (forall x : R, lb <= x <= ub -> comp g f x = id x) -> (forall a : R, lb <= a <= ub -> continuity_pt f a) -> forall b : R, f lb < b < f ub -> continuity_pt g bSublemma:forall x0 y0 z0 : R, Rmax x0 y0 < z0 <-> x0 < z0 /\ y0 < z0x, y, z:RRmin x y > z -> x > z /\ y > zSublemma:forall x0 y0 z0 : R, Rmax x0 y0 < z0 <-> x0 < z0 /\ y0 < z0x, y, z:Rx > z /\ y > z -> Rmin x y > zSublemma:forall x y z : R, Rmax x y < z <-> x < z /\ y < zSublemma2:forall x y z : R, Rmin x y > z <-> x > z /\ y > zforall (f g : R -> R) (lb ub : R), lb < ub -> (forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f y) -> (forall x : R, lb <= x <= ub -> comp g f x = id x) -> (forall a : R, lb <= a <= ub -> continuity_pt f a) -> forall b : R, f lb < b < f ub -> continuity_pt g bSublemma:forall x0 y0 z0 : R, Rmax x0 y0 < z0 <-> x0 < z0 /\ y0 < z0x, y, z:R(if Rle_dec x y then x else y) > z -> x > z /\ y > zSublemma:forall x0 y0 z0 : R, Rmax x0 y0 < z0 <-> x0 < z0 /\ y0 < z0x, y, z:Rx > z /\ y > z -> Rmin x y > zSublemma:forall x y z : R, Rmax x y < z <-> x < z /\ y < zSublemma2:forall x y z : R, Rmin x y > z <-> x > z /\ y > zforall (f g : R -> R) (lb ub : R), lb < ub -> (forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f y) -> (forall x : R, lb <= x <= ub -> comp g f x = id x) -> (forall a : R, lb <= a <= ub -> continuity_pt f a) -> forall b : R, f lb < b < f ub -> continuity_pt g bSublemma:forall x0 y0 z0 : R, Rmax x0 y0 < z0 <-> x0 < z0 /\ y0 < z0x, y, z:RHyp:x <= yHyp2:x > zx > z /\ y > zSublemma:forall x0 y0 z0 : R, Rmax x0 y0 < z0 <-> x0 < z0 /\ y0 < z0x, y, z:RHyp:~ x <= yHyp2:y > zx > z /\ y > zSublemma:forall x0 y0 z0 : R, Rmax x0 y0 < z0 <-> x0 < z0 /\ y0 < z0x, y, z:Rx > z /\ y > z -> Rmin x y > zSublemma:forall x y z : R, Rmax x y < z <-> x < z /\ y < zSublemma2:forall x y z : R, Rmin x y > z <-> x > z /\ y > zforall (f g : R -> R) (lb ub : R), lb < ub -> (forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f y) -> (forall x : R, lb <= x <= ub -> comp g f x = id x) -> (forall a : R, lb <= a <= ub -> continuity_pt f a) -> forall b : R, f lb < b < f ub -> continuity_pt g bSublemma:forall x0 y0 z0 : R, Rmax x0 y0 < z0 <-> x0 < z0 /\ y0 < z0x, y, z:RHyp:x <= yHyp2:x > zx > zSublemma:forall x0 y0 z0 : R, Rmax x0 y0 < z0 <-> x0 < z0 /\ y0 < z0x, y, z:RHyp:x <= yHyp2:x > zy > zSublemma:forall x0 y0 z0 : R, Rmax x0 y0 < z0 <-> x0 < z0 /\ y0 < z0x, y, z:RHyp:~ x <= yHyp2:y > zx > z /\ y > zSublemma:forall x0 y0 z0 : R, Rmax x0 y0 < z0 <-> x0 < z0 /\ y0 < z0x, y, z:Rx > z /\ y > z -> Rmin x y > zSublemma:forall x y z : R, Rmax x y < z <-> x < z /\ y < zSublemma2:forall x y z : R, Rmin x y > z <-> x > z /\ y > zforall (f g : R -> R) (lb ub : R), lb < ub -> (forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f y) -> (forall x : R, lb <= x <= ub -> comp g f x = id x) -> (forall a : R, lb <= a <= ub -> continuity_pt f a) -> forall b : R, f lb < b < f ub -> continuity_pt g bSublemma:forall x0 y0 z0 : R, Rmax x0 y0 < z0 <-> x0 < z0 /\ y0 < z0x, y, z:RHyp:x <= yHyp2:x > zy > zSublemma:forall x0 y0 z0 : R, Rmax x0 y0 < z0 <-> x0 < z0 /\ y0 < z0x, y, z:RHyp:~ x <= yHyp2:y > zx > z /\ y > zSublemma:forall x0 y0 z0 : R, Rmax x0 y0 < z0 <-> x0 < z0 /\ y0 < z0x, y, z:Rx > z /\ y > z -> Rmin x y > zSublemma:forall x y z : R, Rmax x y < z <-> x < z /\ y < zSublemma2:forall x y z : R, Rmin x y > z <-> x > z /\ y > zforall (f g : R -> R) (lb ub : R), lb < ub -> (forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f y) -> (forall x : R, lb <= x <= ub -> comp g f x = id x) -> (forall a : R, lb <= a <= ub -> continuity_pt f a) -> forall b : R, f lb < b < f ub -> continuity_pt g bSublemma:forall x0 y0 z0 : R, Rmax x0 y0 < z0 <-> x0 < z0 /\ y0 < z0x, y, z:RHyp:~ x <= yHyp2:y > zx > z /\ y > zSublemma:forall x0 y0 z0 : R, Rmax x0 y0 < z0 <-> x0 < z0 /\ y0 < z0x, y, z:Rx > z /\ y > z -> Rmin x y > zSublemma:forall x y z : R, Rmax x y < z <-> x < z /\ y < zSublemma2:forall x y z : R, Rmin x y > z <-> x > z /\ y > zforall (f g : R -> R) (lb ub : R), lb < ub -> (forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f y) -> (forall x : R, lb <= x <= ub -> comp g f x = id x) -> (forall a : R, lb <= a <= ub -> continuity_pt f a) -> forall b : R, f lb < b < f ub -> continuity_pt g bSublemma:forall x0 y0 z0 : R, Rmax x0 y0 < z0 <-> x0 < z0 /\ y0 < z0x, y, z:RHyp:~ x <= yHyp2:y > zx > zSublemma:forall x0 y0 z0 : R, Rmax x0 y0 < z0 <-> x0 < z0 /\ y0 < z0x, y, z:RHyp:~ x <= yHyp2:y > zy > zSublemma:forall x0 y0 z0 : R, Rmax x0 y0 < z0 <-> x0 < z0 /\ y0 < z0x, y, z:Rx > z /\ y > z -> Rmin x y > zSublemma:forall x y z : R, Rmax x y < z <-> x < z /\ y < zSublemma2:forall x y z : R, Rmin x y > z <-> x > z /\ y > zforall (f g : R -> R) (lb ub : R), lb < ub -> (forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f y) -> (forall x : R, lb <= x <= ub -> comp g f x = id x) -> (forall a : R, lb <= a <= ub -> continuity_pt f a) -> forall b : R, f lb < b < f ub -> continuity_pt g bSublemma:forall x0 y0 z0 : R, Rmax x0 y0 < z0 <-> x0 < z0 /\ y0 < z0x, y, z:RHyp:~ x <= yHyp2:y > zz < ySublemma:forall x0 y0 z0 : R, Rmax x0 y0 < z0 <-> x0 < z0 /\ y0 < z0x, y, z:RHyp:~ x <= yHyp2:y > zy < xSublemma:forall x0 y0 z0 : R, Rmax x0 y0 < z0 <-> x0 < z0 /\ y0 < z0x, y, z:RHyp:~ x <= yHyp2:y > zy > zSublemma:forall x0 y0 z0 : R, Rmax x0 y0 < z0 <-> x0 < z0 /\ y0 < z0x, y, z:Rx > z /\ y > z -> Rmin x y > zSublemma:forall x y z : R, Rmax x y < z <-> x < z /\ y < zSublemma2:forall x y z : R, Rmin x y > z <-> x > z /\ y > zforall (f g : R -> R) (lb ub : R), lb < ub -> (forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f y) -> (forall x : R, lb <= x <= ub -> comp g f x = id x) -> (forall a : R, lb <= a <= ub -> continuity_pt f a) -> forall b : R, f lb < b < f ub -> continuity_pt g bSublemma:forall x0 y0 z0 : R, Rmax x0 y0 < z0 <-> x0 < z0 /\ y0 < z0x, y, z:RHyp:~ x <= yHyp2:y > zy < xSublemma:forall x0 y0 z0 : R, Rmax x0 y0 < z0 <-> x0 < z0 /\ y0 < z0x, y, z:RHyp:~ x <= yHyp2:y > zy > zSublemma:forall x0 y0 z0 : R, Rmax x0 y0 < z0 <-> x0 < z0 /\ y0 < z0x, y, z:Rx > z /\ y > z -> Rmin x y > zSublemma:forall x y z : R, Rmax x y < z <-> x < z /\ y < zSublemma2:forall x y z : R, Rmin x y > z <-> x > z /\ y > zforall (f g : R -> R) (lb ub : R), lb < ub -> (forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f y) -> (forall x : R, lb <= x <= ub -> comp g f x = id x) -> (forall a : R, lb <= a <= ub -> continuity_pt f a) -> forall b : R, f lb < b < f ub -> continuity_pt g bSublemma:forall x0 y0 z0 : R, Rmax x0 y0 < z0 <-> x0 < z0 /\ y0 < z0x, y, z:RHyp:~ x <= yHyp2:y > zforall x0 y0 : R, ~ x0 <= y0 -> x0 > y0Sublemma:forall x0 y0 z0 : R, Rmax x0 y0 < z0 <-> x0 < z0 /\ y0 < z0x, y, z:RHyp:~ x <= yHyp2:y > zTemp:forall x0 y0 : R, ~ x0 <= y0 -> x0 > y0y < xSublemma:forall x0 y0 z0 : R, Rmax x0 y0 < z0 <-> x0 < z0 /\ y0 < z0x, y, z:RHyp:~ x <= yHyp2:y > zy > zSublemma:forall x0 y0 z0 : R, Rmax x0 y0 < z0 <-> x0 < z0 /\ y0 < z0x, y, z:Rx > z /\ y > z -> Rmin x y > zSublemma:forall x y z : R, Rmax x y < z <-> x < z /\ y < zSublemma2:forall x y z : R, Rmin x y > z <-> x > z /\ y > zforall (f g : R -> R) (lb ub : R), lb < ub -> (forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f y) -> (forall x : R, lb <= x <= ub -> comp g f x = id x) -> (forall a : R, lb <= a <= ub -> continuity_pt f a) -> forall b : R, f lb < b < f ub -> continuity_pt g bSublemma:forall x0 y0 z0 : R, Rmax x0 y0 < z0 <-> x0 < z0 /\ y0 < z0x, y, z:RHyp:~ x <= yHyp2:y > zm, n:RHypmn:~ m <= nm > nSublemma:forall x0 y0 z0 : R, Rmax x0 y0 < z0 <-> x0 < z0 /\ y0 < z0x, y, z:RHyp:~ x <= yHyp2:y > zTemp:forall x0 y0 : R, ~ x0 <= y0 -> x0 > y0y < xSublemma:forall x0 y0 z0 : R, Rmax x0 y0 < z0 <-> x0 < z0 /\ y0 < z0x, y, z:RHyp:~ x <= yHyp2:y > zy > zSublemma:forall x0 y0 z0 : R, Rmax x0 y0 < z0 <-> x0 < z0 /\ y0 < z0x, y, z:Rx > z /\ y > z -> Rmin x y > zSublemma:forall x y z : R, Rmax x y < z <-> x < z /\ y < zSublemma2:forall x y z : R, Rmin x y > z <-> x > z /\ y > zforall (f g : R -> R) (lb ub : R), lb < ub -> (forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f y) -> (forall x : R, lb <= x <= ub -> comp g f x = id x) -> (forall a : R, lb <= a <= ub -> continuity_pt f a) -> forall b : R, f lb < b < f ub -> continuity_pt g bSublemma:forall x0 y0 z0 : R, Rmax x0 y0 < z0 <-> x0 < z0 /\ y0 < z0x, y, z:RHyp:~ x <= yHyp2:y > zTemp:forall x0 y0 : R, ~ x0 <= y0 -> x0 > y0y < xSublemma:forall x0 y0 z0 : R, Rmax x0 y0 < z0 <-> x0 < z0 /\ y0 < z0x, y, z:RHyp:~ x <= yHyp2:y > zy > zSublemma:forall x0 y0 z0 : R, Rmax x0 y0 < z0 <-> x0 < z0 /\ y0 < z0x, y, z:Rx > z /\ y > z -> Rmin x y > zSublemma:forall x y z : R, Rmax x y < z <-> x < z /\ y < zSublemma2:forall x y z : R, Rmin x y > z <-> x > z /\ y > zforall (f g : R -> R) (lb ub : R), lb < ub -> (forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f y) -> (forall x : R, lb <= x <= ub -> comp g f x = id x) -> (forall a : R, lb <= a <= ub -> continuity_pt f a) -> forall b : R, f lb < b < f ub -> continuity_pt g bSublemma:forall x0 y0 z0 : R, Rmax x0 y0 < z0 <-> x0 < z0 /\ y0 < z0x, y, z:RHyp:~ x <= yHyp2:y > zy > zSublemma:forall x0 y0 z0 : R, Rmax x0 y0 < z0 <-> x0 < z0 /\ y0 < z0x, y, z:Rx > z /\ y > z -> Rmin x y > zSublemma:forall x y z : R, Rmax x y < z <-> x < z /\ y < zSublemma2:forall x y z : R, Rmin x y > z <-> x > z /\ y > zforall (f g : R -> R) (lb ub : R), lb < ub -> (forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f y) -> (forall x : R, lb <= x <= ub -> comp g f x = id x) -> (forall a : R, lb <= a <= ub -> continuity_pt f a) -> forall b : R, f lb < b < f ub -> continuity_pt g bSublemma:forall x0 y0 z0 : R, Rmax x0 y0 < z0 <-> x0 < z0 /\ y0 < z0x, y, z:Rx > z /\ y > z -> Rmin x y > zSublemma:forall x y z : R, Rmax x y < z <-> x < z /\ y < zSublemma2:forall x y z : R, Rmin x y > z <-> x > z /\ y > zforall (f g : R -> R) (lb ub : R), lb < ub -> (forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f y) -> (forall x : R, lb <= x <= ub -> comp g f x = id x) -> (forall a : R, lb <= a <= ub -> continuity_pt f a) -> forall b : R, f lb < b < f ub -> continuity_pt g bSublemma:forall x0 y0 z0 : R, Rmax x0 y0 < z0 <-> x0 < z0 /\ y0 < z0x, y, z:RHyp:x > z /\ y > zRmin x y > zSublemma:forall x y z : R, Rmax x y < z <-> x < z /\ y < zSublemma2:forall x y z : R, Rmin x y > z <-> x > z /\ y > zforall (f g : R -> R) (lb ub : R), lb < ub -> (forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f y) -> (forall x : R, lb <= x <= ub -> comp g f x = id x) -> (forall a : R, lb <= a <= ub -> continuity_pt f a) -> forall b : R, f lb < b < f ub -> continuity_pt g bSublemma:forall x0 y0 z0 : R, Rmax x0 y0 < z0 <-> x0 < z0 /\ y0 < z0x, y, z:RHyp:x > z /\ y > z(if Rle_dec x y then x else y) > zSublemma:forall x y z : R, Rmax x y < z <-> x < z /\ y < zSublemma2:forall x y z : R, Rmin x y > z <-> x > z /\ y > zforall (f g : R -> R) (lb ub : R), lb < ub -> (forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f y) -> (forall x : R, lb <= x <= ub -> comp g f x = id x) -> (forall a : R, lb <= a <= ub -> continuity_pt f a) -> forall b : R, f lb < b < f ub -> continuity_pt g bSublemma:forall x0 y0 z0 : R, Rmax x0 y0 < z0 <-> x0 < z0 /\ y0 < z0x, y, z:RHyp:x > z /\ y > zx <= y -> x > zSublemma:forall x0 y0 z0 : R, Rmax x0 y0 < z0 <-> x0 < z0 /\ y0 < z0x, y, z:RHyp:x > z /\ y > z~ x <= y -> y > zSublemma:forall x y z : R, Rmax x y < z <-> x < z /\ y < zSublemma2:forall x y z : R, Rmin x y > z <-> x > z /\ y > zforall (f g : R -> R) (lb ub : R), lb < ub -> (forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f y) -> (forall x : R, lb <= x <= ub -> comp g f x = id x) -> (forall a : R, lb <= a <= ub -> continuity_pt f a) -> forall b : R, f lb < b < f ub -> continuity_pt g bSublemma:forall x0 y0 z0 : R, Rmax x0 y0 < z0 <-> x0 < z0 /\ y0 < z0x, y, z:RHyp:x > z /\ y > z~ x <= y -> y > zSublemma:forall x y z : R, Rmax x y < z <-> x < z /\ y < zSublemma2:forall x y z : R, Rmin x y > z <-> x > z /\ y > zforall (f g : R -> R) (lb ub : R), lb < ub -> (forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f y) -> (forall x : R, lb <= x <= ub -> comp g f x = id x) -> (forall a : R, lb <= a <= ub -> continuity_pt f a) -> forall b : R, f lb < b < f ub -> continuity_pt g bSublemma:forall x y z : R, Rmax x y < z <-> x < z /\ y < zSublemma2:forall x y z : R, Rmin x y > z <-> x > z /\ y > zforall (f g : R -> R) (lb ub : R), lb < ub -> (forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f y) -> (forall x : R, lb <= x <= ub -> comp g f x = id x) -> (forall a : R, lb <= a <= ub -> continuity_pt f a) -> forall b : R, f lb < b < f ub -> continuity_pt g bSublemma:forall x y z : R, Rmax x y < z <-> x < z /\ y < zSublemma2:forall x y z : R, Rmin x y > z <-> x > z /\ y > zforall x y : R, x <= y /\ x <> y -> x < ySublemma:forall x y z : R, Rmax x y < z <-> x < z /\ y < zSublemma2:forall x y z : R, Rmin x y > z <-> x > z /\ y > zSublemma3:forall x y : R, x <= y /\ x <> y -> x < yforall (f g : R -> R) (lb ub : R), lb < ub -> (forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f y) -> (forall x : R, lb <= x <= ub -> comp g f x = id x) -> (forall a : R, lb <= a <= ub -> continuity_pt f a) -> forall b : R, f lb < b < f ub -> continuity_pt g bSublemma:forall x y z : R, Rmax x y < z <-> x < z /\ y < zSublemma2:forall x y z : R, Rmin x y > z <-> x > z /\ y > zm, n:RHyp:m <= n /\ m <> nm < nSublemma:forall x y z : R, Rmax x y < z <-> x < z /\ y < zSublemma2:forall x y z : R, Rmin x y > z <-> x > z /\ y > zSublemma3:forall x y : R, x <= y /\ x <> y -> x < yforall (f g : R -> R) (lb ub : R), lb < ub -> (forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f y) -> (forall x : R, lb <= x <= ub -> comp g f x = id x) -> (forall a : R, lb <= a <= ub -> continuity_pt f a) -> forall b : R, f lb < b < f ub -> continuity_pt g bSublemma:forall x y z : R, Rmax x y < z <-> x < z /\ y < zSublemma2:forall x y z : R, Rmin x y > z <-> x > z /\ y > zm, n:RHyp:(m < n \/ m = n) /\ m <> nm < nSublemma:forall x y z : R, Rmax x y < z <-> x < z /\ y < zSublemma2:forall x y z : R, Rmin x y > z <-> x > z /\ y > zSublemma3:forall x y : R, x <= y /\ x <> y -> x < yforall (f g : R -> R) (lb ub : R), lb < ub -> (forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f y) -> (forall x : R, lb <= x <= ub -> comp g f x = id x) -> (forall a : R, lb <= a <= ub -> continuity_pt f a) -> forall b : R, f lb < b < f ub -> continuity_pt g bSublemma:forall x y z : R, Rmax x y < z <-> x < z /\ y < zSublemma2:forall x y z : R, Rmin x y > z <-> x > z /\ y > zm, n:RHyp1:m < n \/ m = nHyp2:m <> nm < nSublemma:forall x y z : R, Rmax x y < z <-> x < z /\ y < zSublemma2:forall x y z : R, Rmin x y > z <-> x > z /\ y > zSublemma3:forall x y : R, x <= y /\ x <> y -> x < yforall (f g : R -> R) (lb ub : R), lb < ub -> (forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f y) -> (forall x : R, lb <= x <= ub -> comp g f x = id x) -> (forall a : R, lb <= a <= ub -> continuity_pt f a) -> forall b : R, f lb < b < f ub -> continuity_pt g bSublemma:forall x y z : R, Rmax x y < z <-> x < z /\ y < zSublemma2:forall x y z : R, Rmin x y > z <-> x > z /\ y > zm, n:RHyp1:m < n \/ m = nHyp2:m <> nm < n -> m < nSublemma:forall x y z : R, Rmax x y < z <-> x < z /\ y < zSublemma2:forall x y z : R, Rmin x y > z <-> x > z /\ y > zm, n:RHyp1:m < n \/ m = nHyp2:m <> nm = n -> m < nSublemma:forall x y z : R, Rmax x y < z <-> x < z /\ y < zSublemma2:forall x y z : R, Rmin x y > z <-> x > z /\ y > zSublemma3:forall x y : R, x <= y /\ x <> y -> x < yforall (f g : R -> R) (lb ub : R), lb < ub -> (forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f y) -> (forall x : R, lb <= x <= ub -> comp g f x = id x) -> (forall a : R, lb <= a <= ub -> continuity_pt f a) -> forall b : R, f lb < b < f ub -> continuity_pt g bSublemma:forall x y z : R, Rmax x y < z <-> x < z /\ y < zSublemma2:forall x y z : R, Rmin x y > z <-> x > z /\ y > zm, n:RHyp1:m < n \/ m = nHyp2:m <> nm = n -> m < nSublemma:forall x y z : R, Rmax x y < z <-> x < z /\ y < zSublemma2:forall x y z : R, Rmin x y > z <-> x > z /\ y > zSublemma3:forall x y : R, x <= y /\ x <> y -> x < yforall (f g : R -> R) (lb ub : R), lb < ub -> (forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f y) -> (forall x : R, lb <= x <= ub -> comp g f x = id x) -> (forall a : R, lb <= a <= ub -> continuity_pt f a) -> forall b : R, f lb < b < f ub -> continuity_pt g bSublemma:forall x y z : R, Rmax x y < z <-> x < z /\ y < zSublemma2:forall x y z : R, Rmin x y > z <-> x > z /\ y > zSublemma3:forall x y : R, x <= y /\ x <> y -> x < yforall (f g : R -> R) (lb ub : R), lb < ub -> (forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f y) -> (forall x : R, lb <= x <= ub -> comp g f x = id x) -> (forall a : R, lb <= a <= ub -> continuity_pt f a) -> forall b : R, f lb < b < f ub -> continuity_pt g bSublemma:forall x y z : R, Rmax x y < z <-> x < z /\ y < zSublemma2:forall x y z : R, Rmin x y > z <-> x > z /\ y > zSublemma3:forall x y : R, x <= y /\ x <> y -> x < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f yf_eq_g:forall x : R, lb <= x <= ub -> comp g f x = id xf_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubcontinuity_pt g bSublemma:forall x y z : R, Rmax x y < z <-> x < z /\ y < zSublemma2:forall x y z : R, Rmin x y > z <-> x > z /\ y > zSublemma3:forall x y : R, x <= y /\ x <> y -> x < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f yf_eq_g:forall x : R, lb <= x <= ub -> comp g f x = id xf_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubforall x y : R, lb <= x -> x <= y -> y <= ub -> f x <= f ySublemma:forall x y z : R, Rmax x y < z <-> x < z /\ y < zSublemma2:forall x y z : R, Rmin x y > z <-> x > z /\ y > zSublemma3:forall x y : R, x <= y /\ x <> y -> x < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f yf_eq_g:forall x : R, lb <= x <= ub -> comp g f x = id xf_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x y : R, lb <= x -> x <= y -> y <= ub -> f x <= f ycontinuity_pt g bSublemma:forall x y z : R, Rmax x y < z <-> x < z /\ y < zSublemma2:forall x y z : R, Rmin x y > z <-> x > z /\ y > zSublemma3:forall x y : R, x <= y /\ x <> y -> x < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f yf_eq_g:forall x : R, lb <= x <= ub -> comp g f x = id xf_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubm, n:Rcond1:lb <= mcond2:m <= ncond3:n <= ubf m <= f nSublemma:forall x y z : R, Rmax x y < z <-> x < z /\ y < zSublemma2:forall x y z : R, Rmin x y > z <-> x > z /\ y > zSublemma3:forall x y : R, x <= y /\ x <> y -> x < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f yf_eq_g:forall x : R, lb <= x <= ub -> comp g f x = id xf_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x y : R, lb <= x -> x <= y -> y <= ub -> f x <= f ycontinuity_pt g bSublemma:forall x y z : R, Rmax x y < z <-> x < z /\ y < zSublemma2:forall x y z : R, Rmin x y > z <-> x > z /\ y > zSublemma3:forall x y : R, x <= y /\ x <> y -> x < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f yf_eq_g:forall x : R, lb <= x <= ub -> comp g f x = id xf_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubm, n:Rcond1:lb <= mcond2:m <= ncond3:n <= ubm < n -> f m <= f nSublemma:forall x y z : R, Rmax x y < z <-> x < z /\ y < zSublemma2:forall x y z : R, Rmin x y > z <-> x > z /\ y > zSublemma3:forall x y : R, x <= y /\ x <> y -> x < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f yf_eq_g:forall x : R, lb <= x <= ub -> comp g f x = id xf_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubm, n:Rcond1:lb <= mcond2:m <= ncond3:n <= ubm = n -> f m <= f nSublemma:forall x y z : R, Rmax x y < z <-> x < z /\ y < zSublemma2:forall x y z : R, Rmin x y > z <-> x > z /\ y > zSublemma3:forall x y : R, x <= y /\ x <> y -> x < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f yf_eq_g:forall x : R, lb <= x <= ub -> comp g f x = id xf_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x y : R, lb <= x -> x <= y -> y <= ub -> f x <= f ycontinuity_pt g bSublemma:forall x y z : R, Rmax x y < z <-> x < z /\ y < zSublemma2:forall x y z : R, Rmin x y > z <-> x > z /\ y > zSublemma3:forall x y : R, x <= y /\ x <> y -> x < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f yf_eq_g:forall x : R, lb <= x <= ub -> comp g f x = id xf_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubm, n:Rcond1:lb <= mcond2:m <= ncond3:n <= ubcond:m < nf m <= f nSublemma:forall x y z : R, Rmax x y < z <-> x < z /\ y < zSublemma2:forall x y z : R, Rmin x y > z <-> x > z /\ y > zSublemma3:forall x y : R, x <= y /\ x <> y -> x < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f yf_eq_g:forall x : R, lb <= x <= ub -> comp g f x = id xf_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubm, n:Rcond1:lb <= mcond2:m <= ncond3:n <= ubm = n -> f m <= f nSublemma:forall x y z : R, Rmax x y < z <-> x < z /\ y < zSublemma2:forall x y z : R, Rmin x y > z <-> x > z /\ y > zSublemma3:forall x y : R, x <= y /\ x <> y -> x < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f yf_eq_g:forall x : R, lb <= x <= ub -> comp g f x = id xf_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x y : R, lb <= x -> x <= y -> y <= ub -> f x <= f ycontinuity_pt g bSublemma:forall x y z : R, Rmax x y < z <-> x < z /\ y < zSublemma2:forall x y z : R, Rmin x y > z <-> x > z /\ y > zSublemma3:forall x y : R, x <= y /\ x <> y -> x < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f yf_eq_g:forall x : R, lb <= x <= ub -> comp g f x = id xf_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubm, n:Rcond1:lb <= mcond2:m <= ncond3:n <= ubm = n -> f m <= f nSublemma:forall x y z : R, Rmax x y < z <-> x < z /\ y < zSublemma2:forall x y z : R, Rmin x y > z <-> x > z /\ y > zSublemma3:forall x y : R, x <= y /\ x <> y -> x < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f yf_eq_g:forall x : R, lb <= x <= ub -> comp g f x = id xf_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x y : R, lb <= x -> x <= y -> y <= ub -> f x <= f ycontinuity_pt g bSublemma:forall x y z : R, Rmax x y < z <-> x < z /\ y < zSublemma2:forall x y z : R, Rmin x y > z <-> x > z /\ y > zSublemma3:forall x y : R, x <= y /\ x <> y -> x < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f yf_eq_g:forall x : R, lb <= x <= ub -> comp g f x = id xf_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x y : R, lb <= x -> x <= y -> y <= ub -> f x <= f ycontinuity_pt g bSublemma:forall x y z : R, Rmax x y < z <-> x < z /\ y < zSublemma2:forall x y z : R, Rmin x y > z <-> x > z /\ y > zSublemma3:forall x y : R, x <= y /\ x <> y -> x < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f yf_eq_g:forall x : R, lb <= x <= ub -> comp g f x = id xf_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x y : R, lb <= x -> x <= y -> y <= ub -> f x <= f yeps:Reps_pos:eps > 0exists alp : R, alp > 0 /\ (forall x : Base R_met, D_x no_cond b x /\ dist R_met x b < alp -> dist R_met (g x) (g b) < eps)Sublemma:forall x y z : R, Rmax x y < z <-> x < z /\ y < zSublemma2:forall x y z : R, Rmin x y > z <-> x > z /\ y > zSublemma3:forall x y : R, x <= y /\ x <> y -> x < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f yf_eq_g:forall x : R, lb <= x <= ub -> comp g f x = id xf_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x y : R, lb <= x -> x <= y -> y <= ub -> f x <= f yeps:Reps_pos:eps > 0exists alp : R, alp > 0 /\ (forall x : R, D_x no_cond b x /\ Rabs (x - b) < alp -> Rabs (g x - g b) < eps)Sublemma:forall x y z : R, Rmax x y < z <-> x < z /\ y < zSublemma2:forall x y z : R, Rmin x y > z <-> x > z /\ y > zSublemma3:forall x y : R, x <= y /\ x <> y -> x < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f yf_eq_g:forall x : R, lb <= x <= ub -> comp g f x = id xf_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x y : R, lb <= x -> x <= y -> y <= ub -> f x <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubexists alp : R, alp > 0 /\ (forall x : R, D_x no_cond b x /\ Rabs (x - b) < alp -> Rabs (g x - g b) < eps)Sublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:RTemp:lb <= x <= ub /\ f x = bexists alp : R, alp > 0 /\ (forall x0 : R, D_x no_cond b x0 /\ Rabs (x0 - b) < alp -> Rabs (g x0 - g b) < eps)Sublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = bexists alp : R, alp > 0 /\ (forall x0 : R, D_x no_cond b x0 /\ Rabs (x0 - b) < alp -> Rabs (g x0 - g b) < eps)Sublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb < xSublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xexists alp : R, alp > 0 /\ (forall x0 : R, D_x no_cond b x0 /\ Rabs (x0 - b) < alp -> Rabs (g x0 - g b) < eps)Sublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = bx <> lbSublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = bTemp:x <> lblb < xSublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xexists alp : R, alp > 0 /\ (forall x0 : R, D_x no_cond b x0 /\ Rabs (x0 - b) < alp -> Rabs (g x0 - g b) < eps)Sublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = bHfalse:x = lbFalseSublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = bTemp:x <> lblb < xSublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xexists alp : R, alp > 0 /\ (forall x0 : R, D_x no_cond b x0 /\ Rabs (x0 - b) < alp -> Rabs (g x0 - g b) < eps)Sublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = bHfalse:x = lbb = f lbSublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = bHfalse:x = lbTemp':b = f lbFalseSublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = bTemp:x <> lblb < xSublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xexists alp : R, alp > 0 /\ (forall x0 : R, D_x no_cond b x0 /\ Rabs (x0 - b) < alp -> Rabs (g x0 - g b) < eps)Sublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = bHfalse:x = lbTemp':b = f lbFalseSublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = bTemp:x <> lblb < xSublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xexists alp : R, alp > 0 /\ (forall x0 : R, D_x no_cond b x0 /\ Rabs (x0 - b) < alp -> Rabs (g x0 - g b) < eps)Sublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = bHfalse:x = lbTemp':b = f lbb <> f lbSublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = bHfalse:x = lbTemp':b = f lbTemp'':b <> f lbFalseSublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = bTemp:x <> lblb < xSublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xexists alp : R, alp > 0 /\ (forall x0 : R, D_x no_cond b x0 /\ Rabs (x0 - b) < alp -> Rabs (g x0 - g b) < eps)Sublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = bHfalse:x = lbTemp':b = f lbTemp'':b <> f lbFalseSublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = bTemp:x <> lblb < xSublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xexists alp : R, alp > 0 /\ (forall x0 : R, D_x no_cond b x0 /\ Rabs (x0 - b) < alp -> Rabs (g x0 - g b) < eps)Sublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = bTemp:x <> lblb < xSublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xexists alp : R, alp > 0 /\ (forall x0 : R, D_x no_cond b x0 /\ Rabs (x0 - b) < alp -> Rabs (g x0 - g b) < eps)Sublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = bTemp:x <> lblb <= x /\ lb <> xSublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xexists alp : R, alp > 0 /\ (forall x0 : R, D_x no_cond b x0 /\ Rabs (x0 - b) < alp -> Rabs (g x0 - g b) < eps)Sublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = bTemp:x <> lblb <= xSublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = bTemp:x <> lblb <> xSublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xexists alp : R, alp > 0 /\ (forall x0 : R, D_x no_cond b x0 /\ Rabs (x0 - b) < alp -> Rabs (g x0 - g b) < eps)Sublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = bTemp:x <> lblb <> xSublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xexists alp : R, alp > 0 /\ (forall x0 : R, D_x no_cond b x0 /\ Rabs (x0 - b) < alp -> Rabs (g x0 - g b) < eps)Sublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = bTemp:x <> lbforall x0 y : R, x0 <> y <-> y <> x0Sublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = bTemp:x <> lbTemp2:forall x0 y : R, x0 <> y <-> y <> x0lb <> xSublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xexists alp : R, alp > 0 /\ (forall x0 : R, D_x no_cond b x0 /\ Rabs (x0 - b) < alp -> Rabs (g x0 - g b) < eps)Sublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = bTemp:x <> lbm, n:Rm <> n <-> n <> mSublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = bTemp:x <> lbTemp2:forall x0 y : R, x0 <> y <-> y <> x0lb <> xSublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xexists alp : R, alp > 0 /\ (forall x0 : R, D_x no_cond b x0 /\ Rabs (x0 - b) < alp -> Rabs (g x0 - g b) < eps)Sublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = bTemp:x <> lbTemp2:forall x0 y : R, x0 <> y <-> y <> x0lb <> xSublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xexists alp : R, alp > 0 /\ (forall x0 : R, D_x no_cond b x0 /\ Rabs (x0 - b) < alp -> Rabs (g x0 - g b) < eps)Sublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xexists alp : R, alp > 0 /\ (forall x0 : R, D_x no_cond b x0 /\ Rabs (x0 - b) < alp -> Rabs (g x0 - g b) < eps)Sublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx < ubSublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubexists alp : R, alp > 0 /\ (forall x0 : R, D_x no_cond b x0 /\ Rabs (x0 - b) < alp -> Rabs (g x0 - g b) < eps)Sublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx <> ubSublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xTemp:x <> ubx < ubSublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubexists alp : R, alp > 0 /\ (forall x0 : R, D_x no_cond b x0 /\ Rabs (x0 - b) < alp -> Rabs (g x0 - g b) < eps)Sublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xHfalse:x = ubFalseSublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xTemp:x <> ubx < ubSublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubexists alp : R, alp > 0 /\ (forall x0 : R, D_x no_cond b x0 /\ Rabs (x0 - b) < alp -> Rabs (g x0 - g b) < eps)Sublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xHfalse:x = ubb = f ubSublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xHfalse:x = ubTemp':b = f ubFalseSublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xTemp:x <> ubx < ubSublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubexists alp : R, alp > 0 /\ (forall x0 : R, D_x no_cond b x0 /\ Rabs (x0 - b) < alp -> Rabs (g x0 - g b) < eps)Sublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xHfalse:x = ubTemp':b = f ubFalseSublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xTemp:x <> ubx < ubSublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubexists alp : R, alp > 0 /\ (forall x0 : R, D_x no_cond b x0 /\ Rabs (x0 - b) < alp -> Rabs (g x0 - g b) < eps)Sublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xHfalse:x = ubTemp':b = f ubb <> f ubSublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xHfalse:x = ubTemp':b = f ubTemp'':b <> f ubFalseSublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xTemp:x <> ubx < ubSublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubexists alp : R, alp > 0 /\ (forall x0 : R, D_x no_cond b x0 /\ Rabs (x0 - b) < alp -> Rabs (g x0 - g b) < eps)Sublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xHfalse:x = ubTemp':b = f ubTemp'':b <> f ubFalseSublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xTemp:x <> ubx < ubSublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubexists alp : R, alp > 0 /\ (forall x0 : R, D_x no_cond b x0 /\ Rabs (x0 - b) < alp -> Rabs (g x0 - g b) < eps)Sublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xTemp:x <> ubx < ubSublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubexists alp : R, alp > 0 /\ (forall x0 : R, D_x no_cond b x0 /\ Rabs (x0 - b) < alp -> Rabs (g x0 - g b) < eps)Sublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xTemp:x <> ubx <= ub /\ x <> ubSublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubexists alp : R, alp > 0 /\ (forall x0 : R, D_x no_cond b x0 /\ Rabs (x0 - b) < alp -> Rabs (g x0 - g b) < eps)Sublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubexists alp : R, alp > 0 /\ (forall x0 : R, D_x no_cond b x0 /\ Rabs (x0 - b) < alp -> Rabs (g x0 - g b) < eps)Sublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rexists alp : R, alp > 0 /\ (forall x0 : R, D_x no_cond b x0 /\ Rabs (x0 - b) < alp -> Rabs (g x0 - g b) < eps)Sublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:Rexists alp : R, alp > 0 /\ (forall x0 : R, D_x no_cond b x0 /\ Rabs (x0 - b) < alp -> Rabs (g x0 - g b) < eps)Sublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbexists alp : R, alp > 0 /\ (forall x0 : R, D_x no_cond b x0 /\ Rabs (x0 - b) < alp -> Rabs (g x0 - g b) < eps)Sublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubexists alp : R, alp > 0 /\ (forall x0 : R, D_x no_cond b x0 /\ Rabs (x0 - b) < alp -> Rabs (g x0 - g b) < eps)Sublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ublb <= x1 <= ubSublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubexists alp : R, alp > 0 /\ (forall x0 : R, D_x no_cond b x0 /\ Rabs (x0 - b) < alp -> Rabs (g x0 - g b) < eps)Sublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ublb <= x1Sublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1 <= ubSublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubexists alp : R, alp > 0 /\ (forall x0 : R, D_x no_cond b x0 /\ Rabs (x0 - b) < alp -> Rabs (g x0 - g b) < eps)Sublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1 <= ubSublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubexists alp : R, alp > 0 /\ (forall x0 : R, D_x no_cond b x0 /\ Rabs (x0 - b) < alp -> Rabs (g x0 - g b) < eps)Sublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1 < ubSublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubexists alp : R, alp > 0 /\ (forall x0 : R, D_x no_cond b x0 /\ Rabs (x0 - b) < alp -> Rabs (g x0 - g b) < eps)Sublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubRmax (x - eps) lb < ubSublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubexists alp : R, alp > 0 /\ (forall x0 : R, D_x no_cond b x0 /\ Rabs (x0 - b) < alp -> Rabs (g x0 - g b) < eps)Sublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx - eps < ub /\ lb < ubSublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubexists alp : R, alp > 0 /\ (forall x0 : R, D_x no_cond b x0 /\ Rabs (x0 - b) < alp -> Rabs (g x0 - g b) < eps)Sublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx - eps < ubSublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ublb < ubSublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubexists alp : R, alp > 0 /\ (forall x0 : R, D_x no_cond b x0 /\ Rabs (x0 - b) < alp -> Rabs (g x0 - g b) < eps)Sublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ublb < ubSublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubexists alp : R, alp > 0 /\ (forall x0 : R, D_x no_cond b x0 /\ Rabs (x0 - b) < alp -> Rabs (g x0 - g b) < eps)Sublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubexists alp : R, alp > 0 /\ (forall x0 : R, D_x no_cond b x0 /\ Rabs (x0 - b) < alp -> Rabs (g x0 - g b) < eps)Sublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ublb <= x2 <= ubSublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubexists alp : R, alp > 0 /\ (forall x0 : R, D_x no_cond b x0 /\ Rabs (x0 - b) < alp -> Rabs (g x0 - g b) < eps)Sublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ublb <= x2Sublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2 <= ubSublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubexists alp : R, alp > 0 /\ (forall x0 : R, D_x no_cond b x0 /\ Rabs (x0 - b) < alp -> Rabs (g x0 - g b) < eps)Sublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx + eps > lb /\ ub > lbSublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2 <= ubSublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubexists alp : R, alp > 0 /\ (forall x0 : R, D_x no_cond b x0 /\ Rabs (x0 - b) < alp -> Rabs (g x0 - g b) < eps)Sublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx + eps > lbSublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubub > lbSublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2 <= ubSublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubexists alp : R, alp > 0 /\ (forall x0 : R, D_x no_cond b x0 /\ Rabs (x0 - b) < alp -> Rabs (g x0 - g b) < eps)Sublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubub > lbSublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2 <= ubSublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubexists alp : R, alp > 0 /\ (forall x0 : R, D_x no_cond b x0 /\ Rabs (x0 - b) < alp -> Rabs (g x0 - g b) < eps)Sublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2 <= ubSublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubexists alp : R, alp > 0 /\ (forall x0 : R, D_x no_cond b x0 /\ Rabs (x0 - b) < alp -> Rabs (g x0 - g b) < eps)Sublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubexists alp : R, alp > 0 /\ (forall x0 : R, D_x no_cond b x0 /\ Rabs (x0 - b) < alp -> Rabs (g x0 - g b) < eps)Sublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx < x2Sublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2exists alp : R, alp > 0 /\ (forall x0 : R, D_x no_cond b x0 /\ Rabs (x0 - b) < alp -> Rabs (g x0 - g b) < eps)Sublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx < Rmin (x + eps) ubSublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2exists alp : R, alp > 0 /\ (forall x0 : R, D_x no_cond b x0 /\ Rabs (x0 - b) < alp -> Rabs (g x0 - g b) < eps)Sublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubRmin (x + eps) ub > xSublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2exists alp : R, alp > 0 /\ (forall x0 : R, D_x no_cond b x0 /\ Rabs (x0 - b) < alp -> Rabs (g x0 - g b) < eps)Sublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx + eps > x /\ ub > xSublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2exists alp : R, alp > 0 /\ (forall x0 : R, D_x no_cond b x0 /\ Rabs (x0 - b) < alp -> Rabs (g x0 - g b) < eps)Sublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2exists alp : R, alp > 0 /\ (forall x0 : R, D_x no_cond b x0 /\ Rabs (x0 - b) < alp -> Rabs (g x0 - g b) < eps)Sublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1 < xSublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xexists alp : R, alp > 0 /\ (forall x0 : R, D_x no_cond b x0 /\ Rabs (x0 - b) < alp -> Rabs (g x0 - g b) < eps)Sublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2Rmax (x - eps) lb < xSublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xexists alp : R, alp > 0 /\ (forall x0 : R, D_x no_cond b x0 /\ Rabs (x0 - b) < alp -> Rabs (g x0 - g b) < eps)Sublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x - eps < x /\ lb < xSublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xexists alp : R, alp > 0 /\ (forall x0 : R, D_x no_cond b x0 /\ Rabs (x0 - b) < alp -> Rabs (g x0 - g b) < eps)Sublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xexists alp : R, alp > 0 /\ (forall x0 : R, D_x no_cond b x0 /\ Rabs (x0 - b) < alp -> Rabs (g x0 - g b) < eps)Sublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xRmin (f x - f x1) (f x2 - f x) > 0 /\ (forall x0 : R, D_x no_cond b x0 /\ Rabs (x0 - b) < Rmin (f x - f x1) (f x2 - f x) -> Rabs (g x0 - g b) < eps)Sublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xRmin (f x - f x1) (f x2 - f x) > 0Sublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xforall x0 : R, D_x no_cond b x0 /\ Rabs (x0 - b) < Rmin (f x - f x1) (f x2 - f x) -> Rabs (g x0 - g b) < epsSublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xf x > f x1Sublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xf x2 > f xSublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xforall x0 : R, D_x no_cond b x0 /\ Rabs (x0 - b) < Rmin (f x - f x1) (f x2 - f x) -> Rabs (g x0 - g b) < epsSublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xf x2 > f xSublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xforall x0 : R, D_x no_cond b x0 /\ Rabs (x0 - b) < Rmin (f x - f x1) (f x2 - f x) -> Rabs (g x0 - g b) < epsSublemma:forall x0 y z : R, Rmax x0 y < z <-> x0 < z /\ y < zSublemma2:forall x0 y z : R, Rmin x0 y > z <-> x0 > z /\ y > zSublemma3:forall x0 y : R, x0 <= y /\ x0 <> y -> x0 < yf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y : R, lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f yeps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xforall x0 : R, D_x no_cond b x0 /\ Rabs (x0 - b) < Rmin (f x - f x1) (f x2 - f x) -> Rabs (g x0 - g b) < epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:RTemp:D_x no_cond b y /\ Rabs (y - b) < Rmin (f x - f x1) (f x2 - f x)Rabs (g y - g b) < epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - b) < Rmin (f x - f x1) (f x2 - f x)Rabs (g y - g b) < epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Rabs (g y - g b) < epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)forall x0 y0 d1 d2 : R, d1 > 0 -> d2 > 0 -> Rabs (y0 - x0) < Rmin d1 d2 -> x0 - d1 <= y0 <= x0 + d2Sublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Temp:forall x0 y0 d1 d2 : R, d1 > 0 -> d2 > 0 -> Rabs (y0 - x0) < Rmin d1 d2 -> x0 - d1 <= y0 <= x0 + d2Rabs (g y - g b) < epsSublemma:forall x3 y1 z : R, Rmax x3 y1 < z <-> x3 < z /\ y1 < zSublemma2:forall x3 y1 z : R, Rmin x3 y1 > z <-> x3 > z /\ y1 > zSublemma3:forall x3 y1 : R, x3 <= y1 /\ x3 <> y1 -> x3 < y1f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x3 y1 : R, lb <= x3 -> x3 < y1 -> y1 <= ub -> f x3 < f y1f_eq_g:forall x3 : R, lb <= x3 <= ub -> comp g f x3 = id x3f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x3 y1 : R, lb <= x3 -> x3 <= y1 -> y1 <= ub -> f x3 <= f y1eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)x0, y0, d1, d2:RH:d1 > 0H0:d2 > 0H1:Rabs (y0 - x0) < Rmin d1 d2x0 - d1 <= y0 <= x0 + d2Sublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Temp:forall x0 y0 d1 d2 : R, d1 > 0 -> d2 > 0 -> Rabs (y0 - x0) < Rmin d1 d2 -> x0 - d1 <= y0 <= x0 + d2Rabs (g y - g b) < epsSublemma:forall x3 y1 z : R, Rmax x3 y1 < z <-> x3 < z /\ y1 < zSublemma2:forall x3 y1 z : R, Rmin x3 y1 > z <-> x3 > z /\ y1 > zSublemma3:forall x3 y1 : R, x3 <= y1 /\ x3 <> y1 -> x3 < y1f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x3 y1 : R, lb <= x3 -> x3 < y1 -> y1 <= ub -> f x3 < f y1f_eq_g:forall x3 : R, lb <= x3 <= ub -> comp g f x3 = id x3f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x3 y1 : R, lb <= x3 -> x3 <= y1 -> y1 <= ub -> f x3 <= f y1eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)x0, y0, d1, d2:RH:d1 > 0H0:d2 > 0H1:Rabs (y0 - x0) < Rmin d1 d2x0 - d1 <= y0Sublemma:forall x3 y1 z : R, Rmax x3 y1 < z <-> x3 < z /\ y1 < zSublemma2:forall x3 y1 z : R, Rmin x3 y1 > z <-> x3 > z /\ y1 > zSublemma3:forall x3 y1 : R, x3 <= y1 /\ x3 <> y1 -> x3 < y1f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x3 y1 : R, lb <= x3 -> x3 < y1 -> y1 <= ub -> f x3 < f y1f_eq_g:forall x3 : R, lb <= x3 <= ub -> comp g f x3 = id x3f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x3 y1 : R, lb <= x3 -> x3 <= y1 -> y1 <= ub -> f x3 <= f y1eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)x0, y0, d1, d2:RH:d1 > 0H0:d2 > 0H1:Rabs (y0 - x0) < Rmin d1 d2y0 <= x0 + d2Sublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Temp:forall x0 y0 d1 d2 : R, d1 > 0 -> d2 > 0 -> Rabs (y0 - x0) < Rmin d1 d2 -> x0 - d1 <= y0 <= x0 + d2Rabs (g y - g b) < epsSublemma:forall x3 y1 z : R, Rmax x3 y1 < z <-> x3 < z /\ y1 < zSublemma2:forall x3 y1 z : R, Rmin x3 y1 > z <-> x3 > z /\ y1 > zSublemma3:forall x3 y1 : R, x3 <= y1 /\ x3 <> y1 -> x3 < y1f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x3 y1 : R, lb <= x3 -> x3 < y1 -> y1 <= ub -> f x3 < f y1f_eq_g:forall x3 : R, lb <= x3 <= ub -> comp g f x3 = id x3f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x3 y1 : R, lb <= x3 -> x3 <= y1 -> y1 <= ub -> f x3 <= f y1eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)x0, y0, d1, d2:RH:d1 > 0H0:d2 > 0H1:Rabs (y0 - x0) < Rmin d1 d2forall x3 y1 z : R, x3 - y1 <= z -> x3 - z <= y1Sublemma:forall x3 y1 z : R, Rmax x3 y1 < z <-> x3 < z /\ y1 < zSublemma2:forall x3 y1 z : R, Rmin x3 y1 > z <-> x3 > z /\ y1 > zSublemma3:forall x3 y1 : R, x3 <= y1 /\ x3 <> y1 -> x3 < y1f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x3 y1 : R, lb <= x3 -> x3 < y1 -> y1 <= ub -> f x3 < f y1f_eq_g:forall x3 : R, lb <= x3 <= ub -> comp g f x3 = id x3f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x3 y1 : R, lb <= x3 -> x3 <= y1 -> y1 <= ub -> f x3 <= f y1eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)x0, y0, d1, d2:RH:d1 > 0H0:d2 > 0H1:Rabs (y0 - x0) < Rmin d1 d2H10:forall x3 y1 z : R, x3 - y1 <= z -> x3 - z <= y1x0 - d1 <= y0Sublemma:forall x3 y1 z : R, Rmax x3 y1 < z <-> x3 < z /\ y1 < zSublemma2:forall x3 y1 z : R, Rmin x3 y1 > z <-> x3 > z /\ y1 > zSublemma3:forall x3 y1 : R, x3 <= y1 /\ x3 <> y1 -> x3 < y1f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x3 y1 : R, lb <= x3 -> x3 < y1 -> y1 <= ub -> f x3 < f y1f_eq_g:forall x3 : R, lb <= x3 <= ub -> comp g f x3 = id x3f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x3 y1 : R, lb <= x3 -> x3 <= y1 -> y1 <= ub -> f x3 <= f y1eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)x0, y0, d1, d2:RH:d1 > 0H0:d2 > 0H1:Rabs (y0 - x0) < Rmin d1 d2y0 <= x0 + d2Sublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Temp:forall x0 y0 d1 d2 : R, d1 > 0 -> d2 > 0 -> Rabs (y0 - x0) < Rmin d1 d2 -> x0 - d1 <= y0 <= x0 + d2Rabs (g y - g b) < epsSublemma:forall x4 y2 z0 : R, Rmax x4 y2 < z0 <-> x4 < z0 /\ y2 < z0Sublemma2:forall x4 y2 z0 : R, Rmin x4 y2 > z0 <-> x4 > z0 /\ y2 > z0Sublemma3:forall x4 y2 : R, x4 <= y2 /\ (x4 = y2 -> False) -> x4 < y2f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x4 y2 : R, lb <= x4 -> x4 < y2 -> y2 <= ub -> f x4 < f y2f_eq_g:forall x4 : R, lb <= x4 <= ub -> comp g f x4 = id x4f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rf_incr_interv2:forall x4 y2 : R, lb <= x4 -> x4 <= y2 -> y2 <= ub -> f x4 <= f y2eps:Reps_pos:eps > 0x:Rf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)x0, y0, d1, d2:RH:d1 > 0H0:d2 > 0H1:Rabs (y0 - x0) < Rmin d1 d2H2:f lb < bH3:b < f ubH4:f lb <= bH5:b <= f ubH6:lb <= xH7:x <= ubH8:lb <= x1H9:x1 <= ubH10:lb <= x2H11:x2 <= ubx3, y1, z:RH12:x3 - y1 <= zx3 - z <= y1Sublemma:forall x3 y1 z : R, Rmax x3 y1 < z <-> x3 < z /\ y1 < zSublemma2:forall x3 y1 z : R, Rmin x3 y1 > z <-> x3 > z /\ y1 > zSublemma3:forall x3 y1 : R, x3 <= y1 /\ x3 <> y1 -> x3 < y1f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x3 y1 : R, lb <= x3 -> x3 < y1 -> y1 <= ub -> f x3 < f y1f_eq_g:forall x3 : R, lb <= x3 <= ub -> comp g f x3 = id x3f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x3 y1 : R, lb <= x3 -> x3 <= y1 -> y1 <= ub -> f x3 <= f y1eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)x0, y0, d1, d2:RH:d1 > 0H0:d2 > 0H1:Rabs (y0 - x0) < Rmin d1 d2H10:forall x3 y1 z : R, x3 - y1 <= z -> x3 - z <= y1x0 - d1 <= y0Sublemma:forall x3 y1 z : R, Rmax x3 y1 < z <-> x3 < z /\ y1 < zSublemma2:forall x3 y1 z : R, Rmin x3 y1 > z <-> x3 > z /\ y1 > zSublemma3:forall x3 y1 : R, x3 <= y1 /\ x3 <> y1 -> x3 < y1f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x3 y1 : R, lb <= x3 -> x3 < y1 -> y1 <= ub -> f x3 < f y1f_eq_g:forall x3 : R, lb <= x3 <= ub -> comp g f x3 = id x3f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x3 y1 : R, lb <= x3 -> x3 <= y1 -> y1 <= ub -> f x3 <= f y1eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)x0, y0, d1, d2:RH:d1 > 0H0:d2 > 0H1:Rabs (y0 - x0) < Rmin d1 d2y0 <= x0 + d2Sublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Temp:forall x0 y0 d1 d2 : R, d1 > 0 -> d2 > 0 -> Rabs (y0 - x0) < Rmin d1 d2 -> x0 - d1 <= y0 <= x0 + d2Rabs (g y - g b) < epsSublemma:forall x3 y1 z : R, Rmax x3 y1 < z <-> x3 < z /\ y1 < zSublemma2:forall x3 y1 z : R, Rmin x3 y1 > z <-> x3 > z /\ y1 > zSublemma3:forall x3 y1 : R, x3 <= y1 /\ x3 <> y1 -> x3 < y1f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x3 y1 : R, lb <= x3 -> x3 < y1 -> y1 <= ub -> f x3 < f y1f_eq_g:forall x3 : R, lb <= x3 <= ub -> comp g f x3 = id x3f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x3 y1 : R, lb <= x3 -> x3 <= y1 -> y1 <= ub -> f x3 <= f y1eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)x0, y0, d1, d2:RH:d1 > 0H0:d2 > 0H1:Rabs (y0 - x0) < Rmin d1 d2H10:forall x3 y1 z : R, x3 - y1 <= z -> x3 - z <= y1x0 - d1 <= y0Sublemma:forall x3 y1 z : R, Rmax x3 y1 < z <-> x3 < z /\ y1 < zSublemma2:forall x3 y1 z : R, Rmin x3 y1 > z <-> x3 > z /\ y1 > zSublemma3:forall x3 y1 : R, x3 <= y1 /\ x3 <> y1 -> x3 < y1f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x3 y1 : R, lb <= x3 -> x3 < y1 -> y1 <= ub -> f x3 < f y1f_eq_g:forall x3 : R, lb <= x3 <= ub -> comp g f x3 = id x3f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x3 y1 : R, lb <= x3 -> x3 <= y1 -> y1 <= ub -> f x3 <= f y1eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)x0, y0, d1, d2:RH:d1 > 0H0:d2 > 0H1:Rabs (y0 - x0) < Rmin d1 d2y0 <= x0 + d2Sublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Temp:forall x0 y0 d1 d2 : R, d1 > 0 -> d2 > 0 -> Rabs (y0 - x0) < Rmin d1 d2 -> x0 - d1 <= y0 <= x0 + d2Rabs (g y - g b) < epsSublemma:forall x3 y1 z : R, Rmax x3 y1 < z <-> x3 < z /\ y1 < zSublemma2:forall x3 y1 z : R, Rmin x3 y1 > z <-> x3 > z /\ y1 > zSublemma3:forall x3 y1 : R, x3 <= y1 /\ x3 <> y1 -> x3 < y1f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x3 y1 : R, lb <= x3 -> x3 < y1 -> y1 <= ub -> f x3 < f y1f_eq_g:forall x3 : R, lb <= x3 <= ub -> comp g f x3 = id x3f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x3 y1 : R, lb <= x3 -> x3 <= y1 -> y1 <= ub -> f x3 <= f y1eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)x0, y0, d1, d2:RH:d1 > 0H0:d2 > 0H1:Rabs (y0 - x0) < Rmin d1 d2H10:forall x3 y1 z : R, x3 - y1 <= z -> x3 - z <= y1x0 - y0 <= d1Sublemma:forall x3 y1 z : R, Rmax x3 y1 < z <-> x3 < z /\ y1 < zSublemma2:forall x3 y1 z : R, Rmin x3 y1 > z <-> x3 > z /\ y1 > zSublemma3:forall x3 y1 : R, x3 <= y1 /\ x3 <> y1 -> x3 < y1f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x3 y1 : R, lb <= x3 -> x3 < y1 -> y1 <= ub -> f x3 < f y1f_eq_g:forall x3 : R, lb <= x3 <= ub -> comp g f x3 = id x3f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x3 y1 : R, lb <= x3 -> x3 <= y1 -> y1 <= ub -> f x3 <= f y1eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)x0, y0, d1, d2:RH:d1 > 0H0:d2 > 0H1:Rabs (y0 - x0) < Rmin d1 d2y0 <= x0 + d2Sublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Temp:forall x0 y0 d1 d2 : R, d1 > 0 -> d2 > 0 -> Rabs (y0 - x0) < Rmin d1 d2 -> x0 - d1 <= y0 <= x0 + d2Rabs (g y - g b) < epsSublemma:forall x3 y1 z : R, Rmax x3 y1 < z <-> x3 < z /\ y1 < zSublemma2:forall x3 y1 z : R, Rmin x3 y1 > z <-> x3 > z /\ y1 > zSublemma3:forall x3 y1 : R, x3 <= y1 /\ x3 <> y1 -> x3 < y1f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x3 y1 : R, lb <= x3 -> x3 < y1 -> y1 <= ub -> f x3 < f y1f_eq_g:forall x3 : R, lb <= x3 <= ub -> comp g f x3 = id x3f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x3 y1 : R, lb <= x3 -> x3 <= y1 -> y1 <= ub -> f x3 <= f y1eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)x0, y0, d1, d2:RH:d1 > 0H0:d2 > 0H1:Rabs (y0 - x0) < Rmin d1 d2H10:forall x3 y1 z : R, x3 - y1 <= z -> x3 - z <= y1x0 - y0 <= Rabs (y0 - x0)Sublemma:forall x3 y1 z : R, Rmax x3 y1 < z <-> x3 < z /\ y1 < zSublemma2:forall x3 y1 z : R, Rmin x3 y1 > z <-> x3 > z /\ y1 > zSublemma3:forall x3 y1 : R, x3 <= y1 /\ x3 <> y1 -> x3 < y1f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x3 y1 : R, lb <= x3 -> x3 < y1 -> y1 <= ub -> f x3 < f y1f_eq_g:forall x3 : R, lb <= x3 <= ub -> comp g f x3 = id x3f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x3 y1 : R, lb <= x3 -> x3 <= y1 -> y1 <= ub -> f x3 <= f y1eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)x0, y0, d1, d2:RH:d1 > 0H0:d2 > 0H1:Rabs (y0 - x0) < Rmin d1 d2H10:forall x3 y1 z : R, x3 - y1 <= z -> x3 - z <= y1Rabs (y0 - x0) <= d1Sublemma:forall x3 y1 z : R, Rmax x3 y1 < z <-> x3 < z /\ y1 < zSublemma2:forall x3 y1 z : R, Rmin x3 y1 > z <-> x3 > z /\ y1 > zSublemma3:forall x3 y1 : R, x3 <= y1 /\ x3 <> y1 -> x3 < y1f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x3 y1 : R, lb <= x3 -> x3 < y1 -> y1 <= ub -> f x3 < f y1f_eq_g:forall x3 : R, lb <= x3 <= ub -> comp g f x3 = id x3f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x3 y1 : R, lb <= x3 -> x3 <= y1 -> y1 <= ub -> f x3 <= f y1eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)x0, y0, d1, d2:RH:d1 > 0H0:d2 > 0H1:Rabs (y0 - x0) < Rmin d1 d2y0 <= x0 + d2Sublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Temp:forall x0 y0 d1 d2 : R, d1 > 0 -> d2 > 0 -> Rabs (y0 - x0) < Rmin d1 d2 -> x0 - d1 <= y0 <= x0 + d2Rabs (g y - g b) < epsSublemma:forall x3 y1 z : R, Rmax x3 y1 < z <-> x3 < z /\ y1 < zSublemma2:forall x3 y1 z : R, Rmin x3 y1 > z <-> x3 > z /\ y1 > zSublemma3:forall x3 y1 : R, x3 <= y1 /\ x3 <> y1 -> x3 < y1f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x3 y1 : R, lb <= x3 -> x3 < y1 -> y1 <= ub -> f x3 < f y1f_eq_g:forall x3 : R, lb <= x3 <= ub -> comp g f x3 = id x3f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x3 y1 : R, lb <= x3 -> x3 <= y1 -> y1 <= ub -> f x3 <= f y1eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)x0, y0, d1, d2:RH:d1 > 0H0:d2 > 0H1:Rabs (y0 - x0) < Rmin d1 d2H10:forall x3 y1 z : R, x3 - y1 <= z -> x3 - z <= y1x0 - y0 <= Rabs (x0 - y0)Sublemma:forall x3 y1 z : R, Rmax x3 y1 < z <-> x3 < z /\ y1 < zSublemma2:forall x3 y1 z : R, Rmin x3 y1 > z <-> x3 > z /\ y1 > zSublemma3:forall x3 y1 : R, x3 <= y1 /\ x3 <> y1 -> x3 < y1f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x3 y1 : R, lb <= x3 -> x3 < y1 -> y1 <= ub -> f x3 < f y1f_eq_g:forall x3 : R, lb <= x3 <= ub -> comp g f x3 = id x3f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x3 y1 : R, lb <= x3 -> x3 <= y1 -> y1 <= ub -> f x3 <= f y1eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)x0, y0, d1, d2:RH:d1 > 0H0:d2 > 0H1:Rabs (y0 - x0) < Rmin d1 d2H10:forall x3 y1 z : R, x3 - y1 <= z -> x3 - z <= y1Rabs (x0 - y0) = Rabs (y0 - x0)Sublemma:forall x3 y1 z : R, Rmax x3 y1 < z <-> x3 < z /\ y1 < zSublemma2:forall x3 y1 z : R, Rmin x3 y1 > z <-> x3 > z /\ y1 > zSublemma3:forall x3 y1 : R, x3 <= y1 /\ x3 <> y1 -> x3 < y1f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x3 y1 : R, lb <= x3 -> x3 < y1 -> y1 <= ub -> f x3 < f y1f_eq_g:forall x3 : R, lb <= x3 <= ub -> comp g f x3 = id x3f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x3 y1 : R, lb <= x3 -> x3 <= y1 -> y1 <= ub -> f x3 <= f y1eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)x0, y0, d1, d2:RH:d1 > 0H0:d2 > 0H1:Rabs (y0 - x0) < Rmin d1 d2H10:forall x3 y1 z : R, x3 - y1 <= z -> x3 - z <= y1Rabs (y0 - x0) <= d1Sublemma:forall x3 y1 z : R, Rmax x3 y1 < z <-> x3 < z /\ y1 < zSublemma2:forall x3 y1 z : R, Rmin x3 y1 > z <-> x3 > z /\ y1 > zSublemma3:forall x3 y1 : R, x3 <= y1 /\ x3 <> y1 -> x3 < y1f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x3 y1 : R, lb <= x3 -> x3 < y1 -> y1 <= ub -> f x3 < f y1f_eq_g:forall x3 : R, lb <= x3 <= ub -> comp g f x3 = id x3f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x3 y1 : R, lb <= x3 -> x3 <= y1 -> y1 <= ub -> f x3 <= f y1eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)x0, y0, d1, d2:RH:d1 > 0H0:d2 > 0H1:Rabs (y0 - x0) < Rmin d1 d2y0 <= x0 + d2Sublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Temp:forall x0 y0 d1 d2 : R, d1 > 0 -> d2 > 0 -> Rabs (y0 - x0) < Rmin d1 d2 -> x0 - d1 <= y0 <= x0 + d2Rabs (g y - g b) < epsSublemma:forall x3 y1 z : R, Rmax x3 y1 < z <-> x3 < z /\ y1 < zSublemma2:forall x3 y1 z : R, Rmin x3 y1 > z <-> x3 > z /\ y1 > zSublemma3:forall x3 y1 : R, x3 <= y1 /\ x3 <> y1 -> x3 < y1f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x3 y1 : R, lb <= x3 -> x3 < y1 -> y1 <= ub -> f x3 < f y1f_eq_g:forall x3 : R, lb <= x3 <= ub -> comp g f x3 = id x3f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x3 y1 : R, lb <= x3 -> x3 <= y1 -> y1 <= ub -> f x3 <= f y1eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)x0, y0, d1, d2:RH:d1 > 0H0:d2 > 0H1:Rabs (y0 - x0) < Rmin d1 d2H10:forall x3 y1 z : R, x3 - y1 <= z -> x3 - z <= y1Rabs (x0 - y0) = Rabs (y0 - x0)Sublemma:forall x3 y1 z : R, Rmax x3 y1 < z <-> x3 < z /\ y1 < zSublemma2:forall x3 y1 z : R, Rmin x3 y1 > z <-> x3 > z /\ y1 > zSublemma3:forall x3 y1 : R, x3 <= y1 /\ x3 <> y1 -> x3 < y1f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x3 y1 : R, lb <= x3 -> x3 < y1 -> y1 <= ub -> f x3 < f y1f_eq_g:forall x3 : R, lb <= x3 <= ub -> comp g f x3 = id x3f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x3 y1 : R, lb <= x3 -> x3 <= y1 -> y1 <= ub -> f x3 <= f y1eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)x0, y0, d1, d2:RH:d1 > 0H0:d2 > 0H1:Rabs (y0 - x0) < Rmin d1 d2H10:forall x3 y1 z : R, x3 - y1 <= z -> x3 - z <= y1Rabs (y0 - x0) <= d1Sublemma:forall x3 y1 z : R, Rmax x3 y1 < z <-> x3 < z /\ y1 < zSublemma2:forall x3 y1 z : R, Rmin x3 y1 > z <-> x3 > z /\ y1 > zSublemma3:forall x3 y1 : R, x3 <= y1 /\ x3 <> y1 -> x3 < y1f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x3 y1 : R, lb <= x3 -> x3 < y1 -> y1 <= ub -> f x3 < f y1f_eq_g:forall x3 : R, lb <= x3 <= ub -> comp g f x3 = id x3f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x3 y1 : R, lb <= x3 -> x3 <= y1 -> y1 <= ub -> f x3 <= f y1eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)x0, y0, d1, d2:RH:d1 > 0H0:d2 > 0H1:Rabs (y0 - x0) < Rmin d1 d2y0 <= x0 + d2Sublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Temp:forall x0 y0 d1 d2 : R, d1 > 0 -> d2 > 0 -> Rabs (y0 - x0) < Rmin d1 d2 -> x0 - d1 <= y0 <= x0 + d2Rabs (g y - g b) < epsSublemma:forall x3 y1 z : R, Rmax x3 y1 < z <-> x3 < z /\ y1 < zSublemma2:forall x3 y1 z : R, Rmin x3 y1 > z <-> x3 > z /\ y1 > zSublemma3:forall x3 y1 : R, x3 <= y1 /\ x3 <> y1 -> x3 < y1f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x3 y1 : R, lb <= x3 -> x3 < y1 -> y1 <= ub -> f x3 < f y1f_eq_g:forall x3 : R, lb <= x3 <= ub -> comp g f x3 = id x3f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x3 y1 : R, lb <= x3 -> x3 <= y1 -> y1 <= ub -> f x3 <= f y1eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)x0, y0, d1, d2:RH:d1 > 0H0:d2 > 0H1:Rabs (y0 - x0) < Rmin d1 d2H10:forall x3 y1 z : R, x3 - y1 <= z -> x3 - z <= y1Rabs (- (x0 - y0)) = Rabs (y0 - x0)Sublemma:forall x3 y1 z : R, Rmax x3 y1 < z <-> x3 < z /\ y1 < zSublemma2:forall x3 y1 z : R, Rmin x3 y1 > z <-> x3 > z /\ y1 > zSublemma3:forall x3 y1 : R, x3 <= y1 /\ x3 <> y1 -> x3 < y1f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x3 y1 : R, lb <= x3 -> x3 < y1 -> y1 <= ub -> f x3 < f y1f_eq_g:forall x3 : R, lb <= x3 <= ub -> comp g f x3 = id x3f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x3 y1 : R, lb <= x3 -> x3 <= y1 -> y1 <= ub -> f x3 <= f y1eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)x0, y0, d1, d2:RH:d1 > 0H0:d2 > 0H1:Rabs (y0 - x0) < Rmin d1 d2H10:forall x3 y1 z : R, x3 - y1 <= z -> x3 - z <= y1Rabs (y0 - x0) <= d1Sublemma:forall x3 y1 z : R, Rmax x3 y1 < z <-> x3 < z /\ y1 < zSublemma2:forall x3 y1 z : R, Rmin x3 y1 > z <-> x3 > z /\ y1 > zSublemma3:forall x3 y1 : R, x3 <= y1 /\ x3 <> y1 -> x3 < y1f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x3 y1 : R, lb <= x3 -> x3 < y1 -> y1 <= ub -> f x3 < f y1f_eq_g:forall x3 : R, lb <= x3 <= ub -> comp g f x3 = id x3f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x3 y1 : R, lb <= x3 -> x3 <= y1 -> y1 <= ub -> f x3 <= f y1eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)x0, y0, d1, d2:RH:d1 > 0H0:d2 > 0H1:Rabs (y0 - x0) < Rmin d1 d2y0 <= x0 + d2Sublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Temp:forall x0 y0 d1 d2 : R, d1 > 0 -> d2 > 0 -> Rabs (y0 - x0) < Rmin d1 d2 -> x0 - d1 <= y0 <= x0 + d2Rabs (g y - g b) < epsSublemma:forall x3 y1 z : R, Rmax x3 y1 < z <-> x3 < z /\ y1 < zSublemma2:forall x3 y1 z : R, Rmin x3 y1 > z <-> x3 > z /\ y1 > zSublemma3:forall x3 y1 : R, x3 <= y1 /\ x3 <> y1 -> x3 < y1f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x3 y1 : R, lb <= x3 -> x3 < y1 -> y1 <= ub -> f x3 < f y1f_eq_g:forall x3 : R, lb <= x3 <= ub -> comp g f x3 = id x3f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x3 y1 : R, lb <= x3 -> x3 <= y1 -> y1 <= ub -> f x3 <= f y1eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)x0, y0, d1, d2:RH:d1 > 0H0:d2 > 0H1:Rabs (y0 - x0) < Rmin d1 d2H10:forall x3 y1 z : R, x3 - y1 <= z -> x3 - z <= y1Rabs (- x0 + - - y0) = Rabs (y0 + - x0)Sublemma:forall x3 y1 z : R, Rmax x3 y1 < z <-> x3 < z /\ y1 < zSublemma2:forall x3 y1 z : R, Rmin x3 y1 > z <-> x3 > z /\ y1 > zSublemma3:forall x3 y1 : R, x3 <= y1 /\ x3 <> y1 -> x3 < y1f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x3 y1 : R, lb <= x3 -> x3 < y1 -> y1 <= ub -> f x3 < f y1f_eq_g:forall x3 : R, lb <= x3 <= ub -> comp g f x3 = id x3f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x3 y1 : R, lb <= x3 -> x3 <= y1 -> y1 <= ub -> f x3 <= f y1eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)x0, y0, d1, d2:RH:d1 > 0H0:d2 > 0H1:Rabs (y0 - x0) < Rmin d1 d2H10:forall x3 y1 z : R, x3 - y1 <= z -> x3 - z <= y1Rabs (y0 - x0) <= d1Sublemma:forall x3 y1 z : R, Rmax x3 y1 < z <-> x3 < z /\ y1 < zSublemma2:forall x3 y1 z : R, Rmin x3 y1 > z <-> x3 > z /\ y1 > zSublemma3:forall x3 y1 : R, x3 <= y1 /\ x3 <> y1 -> x3 < y1f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x3 y1 : R, lb <= x3 -> x3 < y1 -> y1 <= ub -> f x3 < f y1f_eq_g:forall x3 : R, lb <= x3 <= ub -> comp g f x3 = id x3f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x3 y1 : R, lb <= x3 -> x3 <= y1 -> y1 <= ub -> f x3 <= f y1eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)x0, y0, d1, d2:RH:d1 > 0H0:d2 > 0H1:Rabs (y0 - x0) < Rmin d1 d2y0 <= x0 + d2Sublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Temp:forall x0 y0 d1 d2 : R, d1 > 0 -> d2 > 0 -> Rabs (y0 - x0) < Rmin d1 d2 -> x0 - d1 <= y0 <= x0 + d2Rabs (g y - g b) < epsSublemma:forall x3 y1 z : R, Rmax x3 y1 < z <-> x3 < z /\ y1 < zSublemma2:forall x3 y1 z : R, Rmin x3 y1 > z <-> x3 > z /\ y1 > zSublemma3:forall x3 y1 : R, x3 <= y1 /\ x3 <> y1 -> x3 < y1f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x3 y1 : R, lb <= x3 -> x3 < y1 -> y1 <= ub -> f x3 < f y1f_eq_g:forall x3 : R, lb <= x3 <= ub -> comp g f x3 = id x3f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x3 y1 : R, lb <= x3 -> x3 <= y1 -> y1 <= ub -> f x3 <= f y1eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)x0, y0, d1, d2:RH:d1 > 0H0:d2 > 0H1:Rabs (y0 - x0) < Rmin d1 d2H10:forall x3 y1 z : R, x3 - y1 <= z -> x3 - z <= y1Rabs (- x0 + y0) = Rabs (y0 + - x0)Sublemma:forall x3 y1 z : R, Rmax x3 y1 < z <-> x3 < z /\ y1 < zSublemma2:forall x3 y1 z : R, Rmin x3 y1 > z <-> x3 > z /\ y1 > zSublemma3:forall x3 y1 : R, x3 <= y1 /\ x3 <> y1 -> x3 < y1f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x3 y1 : R, lb <= x3 -> x3 < y1 -> y1 <= ub -> f x3 < f y1f_eq_g:forall x3 : R, lb <= x3 <= ub -> comp g f x3 = id x3f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x3 y1 : R, lb <= x3 -> x3 <= y1 -> y1 <= ub -> f x3 <= f y1eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)x0, y0, d1, d2:RH:d1 > 0H0:d2 > 0H1:Rabs (y0 - x0) < Rmin d1 d2H10:forall x3 y1 z : R, x3 - y1 <= z -> x3 - z <= y1Rabs (y0 - x0) <= d1Sublemma:forall x3 y1 z : R, Rmax x3 y1 < z <-> x3 < z /\ y1 < zSublemma2:forall x3 y1 z : R, Rmin x3 y1 > z <-> x3 > z /\ y1 > zSublemma3:forall x3 y1 : R, x3 <= y1 /\ x3 <> y1 -> x3 < y1f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x3 y1 : R, lb <= x3 -> x3 < y1 -> y1 <= ub -> f x3 < f y1f_eq_g:forall x3 : R, lb <= x3 <= ub -> comp g f x3 = id x3f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x3 y1 : R, lb <= x3 -> x3 <= y1 -> y1 <= ub -> f x3 <= f y1eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)x0, y0, d1, d2:RH:d1 > 0H0:d2 > 0H1:Rabs (y0 - x0) < Rmin d1 d2y0 <= x0 + d2Sublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Temp:forall x0 y0 d1 d2 : R, d1 > 0 -> d2 > 0 -> Rabs (y0 - x0) < Rmin d1 d2 -> x0 - d1 <= y0 <= x0 + d2Rabs (g y - g b) < epsSublemma:forall x3 y1 z : R, Rmax x3 y1 < z <-> x3 < z /\ y1 < zSublemma2:forall x3 y1 z : R, Rmin x3 y1 > z <-> x3 > z /\ y1 > zSublemma3:forall x3 y1 : R, x3 <= y1 /\ x3 <> y1 -> x3 < y1f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x3 y1 : R, lb <= x3 -> x3 < y1 -> y1 <= ub -> f x3 < f y1f_eq_g:forall x3 : R, lb <= x3 <= ub -> comp g f x3 = id x3f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x3 y1 : R, lb <= x3 -> x3 <= y1 -> y1 <= ub -> f x3 <= f y1eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)x0, y0, d1, d2:RH:d1 > 0H0:d2 > 0H1:Rabs (y0 - x0) < Rmin d1 d2H10:forall x3 y1 z : R, x3 - y1 <= z -> x3 - z <= y1Rabs (y0 - x0) <= d1Sublemma:forall x3 y1 z : R, Rmax x3 y1 < z <-> x3 < z /\ y1 < zSublemma2:forall x3 y1 z : R, Rmin x3 y1 > z <-> x3 > z /\ y1 > zSublemma3:forall x3 y1 : R, x3 <= y1 /\ x3 <> y1 -> x3 < y1f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x3 y1 : R, lb <= x3 -> x3 < y1 -> y1 <= ub -> f x3 < f y1f_eq_g:forall x3 : R, lb <= x3 <= ub -> comp g f x3 = id x3f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x3 y1 : R, lb <= x3 -> x3 <= y1 -> y1 <= ub -> f x3 <= f y1eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)x0, y0, d1, d2:RH:d1 > 0H0:d2 > 0H1:Rabs (y0 - x0) < Rmin d1 d2y0 <= x0 + d2Sublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Temp:forall x0 y0 d1 d2 : R, d1 > 0 -> d2 > 0 -> Rabs (y0 - x0) < Rmin d1 d2 -> x0 - d1 <= y0 <= x0 + d2Rabs (g y - g b) < epsSublemma:forall x3 y1 z : R, Rmax x3 y1 < z <-> x3 < z /\ y1 < zSublemma2:forall x3 y1 z : R, Rmin x3 y1 > z <-> x3 > z /\ y1 > zSublemma3:forall x3 y1 : R, x3 <= y1 /\ x3 <> y1 -> x3 < y1f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x3 y1 : R, lb <= x3 -> x3 < y1 -> y1 <= ub -> f x3 < f y1f_eq_g:forall x3 : R, lb <= x3 <= ub -> comp g f x3 = id x3f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x3 y1 : R, lb <= x3 -> x3 <= y1 -> y1 <= ub -> f x3 <= f y1eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)x0, y0, d1, d2:RH:d1 > 0H0:d2 > 0H1:Rabs (y0 - x0) < Rmin d1 d2H10:forall x3 y1 z : R, x3 - y1 <= z -> x3 - z <= y1Rabs (y0 - x0) <= Rmin d1 d2Sublemma:forall x3 y1 z : R, Rmax x3 y1 < z <-> x3 < z /\ y1 < zSublemma2:forall x3 y1 z : R, Rmin x3 y1 > z <-> x3 > z /\ y1 > zSublemma3:forall x3 y1 : R, x3 <= y1 /\ x3 <> y1 -> x3 < y1f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x3 y1 : R, lb <= x3 -> x3 < y1 -> y1 <= ub -> f x3 < f y1f_eq_g:forall x3 : R, lb <= x3 <= ub -> comp g f x3 = id x3f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x3 y1 : R, lb <= x3 -> x3 <= y1 -> y1 <= ub -> f x3 <= f y1eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)x0, y0, d1, d2:RH:d1 > 0H0:d2 > 0H1:Rabs (y0 - x0) < Rmin d1 d2H10:forall x3 y1 z : R, x3 - y1 <= z -> x3 - z <= y1Rmin d1 d2 <= d1Sublemma:forall x3 y1 z : R, Rmax x3 y1 < z <-> x3 < z /\ y1 < zSublemma2:forall x3 y1 z : R, Rmin x3 y1 > z <-> x3 > z /\ y1 > zSublemma3:forall x3 y1 : R, x3 <= y1 /\ x3 <> y1 -> x3 < y1f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x3 y1 : R, lb <= x3 -> x3 < y1 -> y1 <= ub -> f x3 < f y1f_eq_g:forall x3 : R, lb <= x3 <= ub -> comp g f x3 = id x3f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x3 y1 : R, lb <= x3 -> x3 <= y1 -> y1 <= ub -> f x3 <= f y1eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)x0, y0, d1, d2:RH:d1 > 0H0:d2 > 0H1:Rabs (y0 - x0) < Rmin d1 d2y0 <= x0 + d2Sublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Temp:forall x0 y0 d1 d2 : R, d1 > 0 -> d2 > 0 -> Rabs (y0 - x0) < Rmin d1 d2 -> x0 - d1 <= y0 <= x0 + d2Rabs (g y - g b) < epsSublemma:forall x3 y1 z : R, Rmax x3 y1 < z <-> x3 < z /\ y1 < zSublemma2:forall x3 y1 z : R, Rmin x3 y1 > z <-> x3 > z /\ y1 > zSublemma3:forall x3 y1 : R, x3 <= y1 /\ x3 <> y1 -> x3 < y1f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x3 y1 : R, lb <= x3 -> x3 < y1 -> y1 <= ub -> f x3 < f y1f_eq_g:forall x3 : R, lb <= x3 <= ub -> comp g f x3 = id x3f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x3 y1 : R, lb <= x3 -> x3 <= y1 -> y1 <= ub -> f x3 <= f y1eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)x0, y0, d1, d2:RH:d1 > 0H0:d2 > 0H1:Rabs (y0 - x0) < Rmin d1 d2H10:forall x3 y1 z : R, x3 - y1 <= z -> x3 - z <= y1Rmin d1 d2 <= d1Sublemma:forall x3 y1 z : R, Rmax x3 y1 < z <-> x3 < z /\ y1 < zSublemma2:forall x3 y1 z : R, Rmin x3 y1 > z <-> x3 > z /\ y1 > zSublemma3:forall x3 y1 : R, x3 <= y1 /\ x3 <> y1 -> x3 < y1f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x3 y1 : R, lb <= x3 -> x3 < y1 -> y1 <= ub -> f x3 < f y1f_eq_g:forall x3 : R, lb <= x3 <= ub -> comp g f x3 = id x3f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x3 y1 : R, lb <= x3 -> x3 <= y1 -> y1 <= ub -> f x3 <= f y1eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)x0, y0, d1, d2:RH:d1 > 0H0:d2 > 0H1:Rabs (y0 - x0) < Rmin d1 d2y0 <= x0 + d2Sublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Temp:forall x0 y0 d1 d2 : R, d1 > 0 -> d2 > 0 -> Rabs (y0 - x0) < Rmin d1 d2 -> x0 - d1 <= y0 <= x0 + d2Rabs (g y - g b) < epsSublemma:forall x3 y1 z : R, Rmax x3 y1 < z <-> x3 < z /\ y1 < zSublemma2:forall x3 y1 z : R, Rmin x3 y1 > z <-> x3 > z /\ y1 > zSublemma3:forall x3 y1 : R, x3 <= y1 /\ x3 <> y1 -> x3 < y1f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x3 y1 : R, lb <= x3 -> x3 < y1 -> y1 <= ub -> f x3 < f y1f_eq_g:forall x3 : R, lb <= x3 <= ub -> comp g f x3 = id x3f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x3 y1 : R, lb <= x3 -> x3 <= y1 -> y1 <= ub -> f x3 <= f y1eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)x0, y0, d1, d2:RH:d1 > 0H0:d2 > 0H1:Rabs (y0 - x0) < Rmin d1 d2y0 <= x0 + d2Sublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Temp:forall x0 y0 d1 d2 : R, d1 > 0 -> d2 > 0 -> Rabs (y0 - x0) < Rmin d1 d2 -> x0 - d1 <= y0 <= x0 + d2Rabs (g y - g b) < epsSublemma:forall x3 y1 z : R, Rmax x3 y1 < z <-> x3 < z /\ y1 < zSublemma2:forall x3 y1 z : R, Rmin x3 y1 > z <-> x3 > z /\ y1 > zSublemma3:forall x3 y1 : R, x3 <= y1 /\ x3 <> y1 -> x3 < y1f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x3 y1 : R, lb <= x3 -> x3 < y1 -> y1 <= ub -> f x3 < f y1f_eq_g:forall x3 : R, lb <= x3 <= ub -> comp g f x3 = id x3f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x3 y1 : R, lb <= x3 -> x3 <= y1 -> y1 <= ub -> f x3 <= f y1eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)x0, y0, d1, d2:RH:d1 > 0H0:d2 > 0H1:Rabs (y0 - x0) < Rmin d1 d2forall x3 y1 z : R, x3 - y1 <= z -> x3 <= y1 + zSublemma:forall x3 y1 z : R, Rmax x3 y1 < z <-> x3 < z /\ y1 < zSublemma2:forall x3 y1 z : R, Rmin x3 y1 > z <-> x3 > z /\ y1 > zSublemma3:forall x3 y1 : R, x3 <= y1 /\ x3 <> y1 -> x3 < y1f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x3 y1 : R, lb <= x3 -> x3 < y1 -> y1 <= ub -> f x3 < f y1f_eq_g:forall x3 : R, lb <= x3 <= ub -> comp g f x3 = id x3f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x3 y1 : R, lb <= x3 -> x3 <= y1 -> y1 <= ub -> f x3 <= f y1eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)x0, y0, d1, d2:RH:d1 > 0H0:d2 > 0H1:Rabs (y0 - x0) < Rmin d1 d2H10:forall x3 y1 z : R, x3 - y1 <= z -> x3 <= y1 + zy0 <= x0 + d2Sublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Temp:forall x0 y0 d1 d2 : R, d1 > 0 -> d2 > 0 -> Rabs (y0 - x0) < Rmin d1 d2 -> x0 - d1 <= y0 <= x0 + d2Rabs (g y - g b) < epsSublemma:forall x4 y2 z0 : R, Rmax x4 y2 < z0 <-> x4 < z0 /\ y2 < z0Sublemma2:forall x4 y2 z0 : R, Rmin x4 y2 > z0 <-> x4 > z0 /\ y2 > z0Sublemma3:forall x4 y2 : R, x4 <= y2 /\ (x4 = y2 -> False) -> x4 < y2f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x4 y2 : R, lb <= x4 -> x4 < y2 -> y2 <= ub -> f x4 < f y2f_eq_g:forall x4 : R, lb <= x4 <= ub -> comp g f x4 = id x4f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rf_incr_interv2:forall x4 y2 : R, lb <= x4 -> x4 <= y2 -> y2 <= ub -> f x4 <= f y2eps:Reps_pos:eps > 0x:Rf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)x0, y0, d1, d2:RH:d1 > 0H0:d2 > 0H1:Rabs (y0 - x0) < Rmin d1 d2H2:f lb < bH3:b < f ubH4:f lb <= bH5:b <= f ubH6:lb <= xH7:x <= ubH8:lb <= x1H9:x1 <= ubH10:lb <= x2H11:x2 <= ubx3, y1, z:RH12:x3 - y1 <= zx3 <= y1 + zSublemma:forall x3 y1 z : R, Rmax x3 y1 < z <-> x3 < z /\ y1 < zSublemma2:forall x3 y1 z : R, Rmin x3 y1 > z <-> x3 > z /\ y1 > zSublemma3:forall x3 y1 : R, x3 <= y1 /\ x3 <> y1 -> x3 < y1f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x3 y1 : R, lb <= x3 -> x3 < y1 -> y1 <= ub -> f x3 < f y1f_eq_g:forall x3 : R, lb <= x3 <= ub -> comp g f x3 = id x3f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x3 y1 : R, lb <= x3 -> x3 <= y1 -> y1 <= ub -> f x3 <= f y1eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)x0, y0, d1, d2:RH:d1 > 0H0:d2 > 0H1:Rabs (y0 - x0) < Rmin d1 d2H10:forall x3 y1 z : R, x3 - y1 <= z -> x3 <= y1 + zy0 <= x0 + d2Sublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Temp:forall x0 y0 d1 d2 : R, d1 > 0 -> d2 > 0 -> Rabs (y0 - x0) < Rmin d1 d2 -> x0 - d1 <= y0 <= x0 + d2Rabs (g y - g b) < epsSublemma:forall x3 y1 z : R, Rmax x3 y1 < z <-> x3 < z /\ y1 < zSublemma2:forall x3 y1 z : R, Rmin x3 y1 > z <-> x3 > z /\ y1 > zSublemma3:forall x3 y1 : R, x3 <= y1 /\ x3 <> y1 -> x3 < y1f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x3 y1 : R, lb <= x3 -> x3 < y1 -> y1 <= ub -> f x3 < f y1f_eq_g:forall x3 : R, lb <= x3 <= ub -> comp g f x3 = id x3f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x3 y1 : R, lb <= x3 -> x3 <= y1 -> y1 <= ub -> f x3 <= f y1eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)x0, y0, d1, d2:RH:d1 > 0H0:d2 > 0H1:Rabs (y0 - x0) < Rmin d1 d2H10:forall x3 y1 z : R, x3 - y1 <= z -> x3 <= y1 + zy0 <= x0 + d2Sublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Temp:forall x0 y0 d1 d2 : R, d1 > 0 -> d2 > 0 -> Rabs (y0 - x0) < Rmin d1 d2 -> x0 - d1 <= y0 <= x0 + d2Rabs (g y - g b) < epsSublemma:forall x3 y1 z : R, Rmax x3 y1 < z <-> x3 < z /\ y1 < zSublemma2:forall x3 y1 z : R, Rmin x3 y1 > z <-> x3 > z /\ y1 > zSublemma3:forall x3 y1 : R, x3 <= y1 /\ x3 <> y1 -> x3 < y1f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x3 y1 : R, lb <= x3 -> x3 < y1 -> y1 <= ub -> f x3 < f y1f_eq_g:forall x3 : R, lb <= x3 <= ub -> comp g f x3 = id x3f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x3 y1 : R, lb <= x3 -> x3 <= y1 -> y1 <= ub -> f x3 <= f y1eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)x0, y0, d1, d2:RH:d1 > 0H0:d2 > 0H1:Rabs (y0 - x0) < Rmin d1 d2H10:forall x3 y1 z : R, x3 - y1 <= z -> x3 <= y1 + zy0 - x0 <= d2Sublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Temp:forall x0 y0 d1 d2 : R, d1 > 0 -> d2 > 0 -> Rabs (y0 - x0) < Rmin d1 d2 -> x0 - d1 <= y0 <= x0 + d2Rabs (g y - g b) < epsSublemma:forall x3 y1 z : R, Rmax x3 y1 < z <-> x3 < z /\ y1 < zSublemma2:forall x3 y1 z : R, Rmin x3 y1 > z <-> x3 > z /\ y1 > zSublemma3:forall x3 y1 : R, x3 <= y1 /\ x3 <> y1 -> x3 < y1f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x3 y1 : R, lb <= x3 -> x3 < y1 -> y1 <= ub -> f x3 < f y1f_eq_g:forall x3 : R, lb <= x3 <= ub -> comp g f x3 = id x3f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x3 y1 : R, lb <= x3 -> x3 <= y1 -> y1 <= ub -> f x3 <= f y1eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)x0, y0, d1, d2:RH:d1 > 0H0:d2 > 0H1:Rabs (y0 - x0) < Rmin d1 d2H10:forall x3 y1 z : R, x3 - y1 <= z -> x3 <= y1 + zy0 - x0 <= Rabs (y0 - x0)Sublemma:forall x3 y1 z : R, Rmax x3 y1 < z <-> x3 < z /\ y1 < zSublemma2:forall x3 y1 z : R, Rmin x3 y1 > z <-> x3 > z /\ y1 > zSublemma3:forall x3 y1 : R, x3 <= y1 /\ x3 <> y1 -> x3 < y1f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x3 y1 : R, lb <= x3 -> x3 < y1 -> y1 <= ub -> f x3 < f y1f_eq_g:forall x3 : R, lb <= x3 <= ub -> comp g f x3 = id x3f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x3 y1 : R, lb <= x3 -> x3 <= y1 -> y1 <= ub -> f x3 <= f y1eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)x0, y0, d1, d2:RH:d1 > 0H0:d2 > 0H1:Rabs (y0 - x0) < Rmin d1 d2H10:forall x3 y1 z : R, x3 - y1 <= z -> x3 <= y1 + zRabs (y0 - x0) <= d2Sublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Temp:forall x0 y0 d1 d2 : R, d1 > 0 -> d2 > 0 -> Rabs (y0 - x0) < Rmin d1 d2 -> x0 - d1 <= y0 <= x0 + d2Rabs (g y - g b) < epsSublemma:forall x3 y1 z : R, Rmax x3 y1 < z <-> x3 < z /\ y1 < zSublemma2:forall x3 y1 z : R, Rmin x3 y1 > z <-> x3 > z /\ y1 > zSublemma3:forall x3 y1 : R, x3 <= y1 /\ x3 <> y1 -> x3 < y1f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x3 y1 : R, lb <= x3 -> x3 < y1 -> y1 <= ub -> f x3 < f y1f_eq_g:forall x3 : R, lb <= x3 <= ub -> comp g f x3 = id x3f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x3 y1 : R, lb <= x3 -> x3 <= y1 -> y1 <= ub -> f x3 <= f y1eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)x0, y0, d1, d2:RH:d1 > 0H0:d2 > 0H1:Rabs (y0 - x0) < Rmin d1 d2H10:forall x3 y1 z : R, x3 - y1 <= z -> x3 <= y1 + zRabs (y0 - x0) <= d2Sublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Temp:forall x0 y0 d1 d2 : R, d1 > 0 -> d2 > 0 -> Rabs (y0 - x0) < Rmin d1 d2 -> x0 - d1 <= y0 <= x0 + d2Rabs (g y - g b) < epsSublemma:forall x3 y1 z : R, Rmax x3 y1 < z <-> x3 < z /\ y1 < zSublemma2:forall x3 y1 z : R, Rmin x3 y1 > z <-> x3 > z /\ y1 > zSublemma3:forall x3 y1 : R, x3 <= y1 /\ x3 <> y1 -> x3 < y1f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x3 y1 : R, lb <= x3 -> x3 < y1 -> y1 <= ub -> f x3 < f y1f_eq_g:forall x3 : R, lb <= x3 <= ub -> comp g f x3 = id x3f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x3 y1 : R, lb <= x3 -> x3 <= y1 -> y1 <= ub -> f x3 <= f y1eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)x0, y0, d1, d2:RH:d1 > 0H0:d2 > 0H1:Rabs (y0 - x0) < Rmin d1 d2H10:forall x3 y1 z : R, x3 - y1 <= z -> x3 <= y1 + zRabs (y0 - x0) <= Rmin d1 d2Sublemma:forall x3 y1 z : R, Rmax x3 y1 < z <-> x3 < z /\ y1 < zSublemma2:forall x3 y1 z : R, Rmin x3 y1 > z <-> x3 > z /\ y1 > zSublemma3:forall x3 y1 : R, x3 <= y1 /\ x3 <> y1 -> x3 < y1f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x3 y1 : R, lb <= x3 -> x3 < y1 -> y1 <= ub -> f x3 < f y1f_eq_g:forall x3 : R, lb <= x3 <= ub -> comp g f x3 = id x3f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x3 y1 : R, lb <= x3 -> x3 <= y1 -> y1 <= ub -> f x3 <= f y1eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)x0, y0, d1, d2:RH:d1 > 0H0:d2 > 0H1:Rabs (y0 - x0) < Rmin d1 d2H10:forall x3 y1 z : R, x3 - y1 <= z -> x3 <= y1 + zRmin d1 d2 <= d2Sublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Temp:forall x0 y0 d1 d2 : R, d1 > 0 -> d2 > 0 -> Rabs (y0 - x0) < Rmin d1 d2 -> x0 - d1 <= y0 <= x0 + d2Rabs (g y - g b) < epsSublemma:forall x3 y1 z : R, Rmax x3 y1 < z <-> x3 < z /\ y1 < zSublemma2:forall x3 y1 z : R, Rmin x3 y1 > z <-> x3 > z /\ y1 > zSublemma3:forall x3 y1 : R, x3 <= y1 /\ x3 <> y1 -> x3 < y1f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x3 y1 : R, lb <= x3 -> x3 < y1 -> y1 <= ub -> f x3 < f y1f_eq_g:forall x3 : R, lb <= x3 <= ub -> comp g f x3 = id x3f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x3 y1 : R, lb <= x3 -> x3 <= y1 -> y1 <= ub -> f x3 <= f y1eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)x0, y0, d1, d2:RH:d1 > 0H0:d2 > 0H1:Rabs (y0 - x0) < Rmin d1 d2H10:forall x3 y1 z : R, x3 - y1 <= z -> x3 <= y1 + zRmin d1 d2 <= d2Sublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Temp:forall x0 y0 d1 d2 : R, d1 > 0 -> d2 > 0 -> Rabs (y0 - x0) < Rmin d1 d2 -> x0 - d1 <= y0 <= x0 + d2Rabs (g y - g b) < epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Temp:forall x0 y0 d1 d2 : R, d1 > 0 -> d2 > 0 -> Rabs (y0 - x0) < Rmin d1 d2 -> x0 - d1 <= y0 <= x0 + d2Rabs (g y - g b) < epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Temp:forall x0 y0 d1 d2 : R, d1 > 0 -> d2 > 0 -> Rabs (y0 - x0) < Rmin d1 d2 -> x0 - d1 <= y0 <= x0 + d2Temp':f x - f x1 > 0 -> f x2 - f x > 0 -> Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x) -> f x - (f x - f x1) <= y <= f x + (f x2 - f x)Rabs (g y - g b) < epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Temp:forall x0 y0 d1 d2 : R, d1 > 0 -> d2 > 0 -> Rabs (y0 - x0) < Rmin d1 d2 -> x0 - d1 <= y0 <= x0 + d2Temp':f x - f x1 > 0 -> f x2 - f x > 0 -> Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x) -> f x1 <= y <= f x + (f x2 - f x)Rabs (g y - g b) < epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Temp:forall x0 y0 d1 d2 : R, d1 > 0 -> d2 > 0 -> Rabs (y0 - x0) < Rmin d1 d2 -> x0 - d1 <= y0 <= x0 + d2Temp':f x - f x1 > 0 -> f x2 - f x > 0 -> Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x) -> f x1 <= y <= f x2Rabs (g y - g b) < epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Temp:forall x0 y0 d1 d2 : R, d1 > 0 -> d2 > 0 -> Rabs (y0 - x0) < Rmin d1 d2 -> x0 - d1 <= y0 <= x0 + d2Temp':f x - f x1 > 0 -> f x2 - f x > 0 -> Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x) -> f x1 <= y <= f x2f x - f x1 > 0Sublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Temp:forall x0 y0 d1 d2 : R, d1 > 0 -> d2 > 0 -> Rabs (y0 - x0) < Rmin d1 d2 -> x0 - d1 <= y0 <= x0 + d2Temp':f x - f x1 > 0 -> f x2 - f x > 0 -> Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x) -> f x1 <= y <= f x2T:f x - f x1 > 0Rabs (g y - g b) < epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Temp:forall x0 y0 d1 d2 : R, d1 > 0 -> d2 > 0 -> Rabs (y0 - x0) < Rmin d1 d2 -> x0 - d1 <= y0 <= x0 + d2Temp':f x - f x1 > 0 -> f x2 - f x > 0 -> Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x) -> f x1 <= y <= f x2f x > f x1Sublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Temp:forall x0 y0 d1 d2 : R, d1 > 0 -> d2 > 0 -> Rabs (y0 - x0) < Rmin d1 d2 -> x0 - d1 <= y0 <= x0 + d2Temp':f x - f x1 > 0 -> f x2 - f x > 0 -> Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x) -> f x1 <= y <= f x2T:f x - f x1 > 0Rabs (g y - g b) < epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Temp:forall x0 y0 d1 d2 : R, d1 > 0 -> d2 > 0 -> Rabs (y0 - x0) < Rmin d1 d2 -> x0 - d1 <= y0 <= x0 + d2Temp':f x - f x1 > 0 -> f x2 - f x > 0 -> Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x) -> f x1 <= y <= f x2T:f x - f x1 > 0Rabs (g y - g b) < epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Temp:forall x0 y0 d1 d2 : R, d1 > 0 -> d2 > 0 -> Rabs (y0 - x0) < Rmin d1 d2 -> x0 - d1 <= y0 <= x0 + d2Temp':f x - f x1 > 0 -> f x2 - f x > 0 -> Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x) -> f x1 <= y <= f x2T:f x - f x1 > 0f x2 - f x > 0Sublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Temp:forall x0 y0 d1 d2 : R, d1 > 0 -> d2 > 0 -> Rabs (y0 - x0) < Rmin d1 d2 -> x0 - d1 <= y0 <= x0 + d2Temp':f x - f x1 > 0 -> f x2 - f x > 0 -> Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x) -> f x1 <= y <= f x2T:f x - f x1 > 0T':f x2 - f x > 0Rabs (g y - g b) < epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Temp:forall x0 y0 d1 d2 : R, d1 > 0 -> d2 > 0 -> Rabs (y0 - x0) < Rmin d1 d2 -> x0 - d1 <= y0 <= x0 + d2Temp':f x - f x1 > 0 -> f x2 - f x > 0 -> Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x) -> f x1 <= y <= f x2T:f x - f x1 > 0f x2 > f xSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Temp:forall x0 y0 d1 d2 : R, d1 > 0 -> d2 > 0 -> Rabs (y0 - x0) < Rmin d1 d2 -> x0 - d1 <= y0 <= x0 + d2Temp':f x - f x1 > 0 -> f x2 - f x > 0 -> Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x) -> f x1 <= y <= f x2T:f x - f x1 > 0T':f x2 - f x > 0Rabs (g y - g b) < epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Temp:forall x0 y0 d1 d2 : R, d1 > 0 -> d2 > 0 -> Rabs (y0 - x0) < Rmin d1 d2 -> x0 - d1 <= y0 <= x0 + d2Temp':f x - f x1 > 0 -> f x2 - f x > 0 -> Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x) -> f x1 <= y <= f x2T:f x - f x1 > 0T':f x2 - f x > 0Rabs (g y - g b) < epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Temp:forall x0 y0 d1 d2 : R, d1 > 0 -> d2 > 0 -> Rabs (y0 - x0) < Rmin d1 d2 -> x0 - d1 <= y0 <= x0 + d2Temp':f x - f x1 > 0 -> f x2 - f x > 0 -> Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x) -> f x1 <= y <= f x2T:f x - f x1 > 0T':f x2 - f x > 0Main:f x1 <= y <= f x2Rabs (g y - g b) < epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2Rabs (g y - g b) < epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1 < x2Sublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2Rabs (g y - g b) < epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2Rabs (g y - g b) < epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2forall a : R, x1 <= a <= x2 -> continuity_pt f aSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f aRabs (g y - g b) < epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a0 : R, lb <= a0 <= ub -> continuity_pt f a0b:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2a:RH:x1 <= a <= x2lb <= aSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a0 : R, lb <= a0 <= ub -> continuity_pt f a0b:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2a:RH:x1 <= a <= x2a <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f aRabs (g y - g b) < epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a0 : R, lb <= a0 <= ub -> continuity_pt f a0b:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2a:RH:x1 <= a <= x2a <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f aRabs (g y - g b) < epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f aRabs (g y - g b) < epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':RTemp:x1 <= x' <= x2 /\ f x' = yRabs (g y - g b) < epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yRabs (g y - g b) < epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yRabs (g (f x') - g (f x)) < epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yRabs (g (f x') - g (f x)) < epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yRabs (id x' - g (f x)) < epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x' <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yRabs (id x' - id x) < epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x' <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yRabs (x' - x) < epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x' <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx - eps <= x' <= x + epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsRabs (x' - x) < epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x' <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx - eps <= x'Sublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx' <= x + epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsRabs (x' - x) < epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x' <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx' <= x + epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsRabs (x' - x) < epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x' <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsRabs (x' - x) < epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x' <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1 < x'Sublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'Rabs (x' - x) < epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x' <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1 <= x' /\ x1 <> x'Sublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'Rabs (x' - x) < epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x' <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1 <> x'Sublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_neq_x':x1 <> x'x1 <= x' /\ x1 <> x'Sublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'Rabs (x' - x) < epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x' <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsHfalse:x1 = x'FalseSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_neq_x':x1 <> x'x1 <= x' /\ x1 <> x'Sublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'Rabs (x' - x) < epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x' <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy, x':Ry_cond:Rabs (y - f x) < Rmin (f x - y) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsHfalse:x1 = x'FalseSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_neq_x':x1 <> x'x1 <= x' /\ x1 <> x'Sublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'Rabs (x' - x) < epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x' <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy, x':Ry_cond:Rabs (y - f x) < Rmin (f x - y) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsHfalse:x1 = x'Rabs (y - f x) < f x - ySublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy, x':Ry_cond:Rabs (y - f x) < Rmin (f x - y) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsHfalse:x1 = x'Hf:Rabs (y - f x) < f x - yFalseSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_neq_x':x1 <> x'x1 <= x' /\ x1 <> x'Sublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'Rabs (x' - x) < epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x' <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy, x':Ry_cond:Rabs (y - f x) < Rmin (f x - y) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsHfalse:x1 = x'Rabs (y - f x) < Rmin (f x - y) (f x2 - f x)Sublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy, x':Ry_cond:Rabs (y - f x) < Rmin (f x - y) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsHfalse:x1 = x'Rmin (f x - y) (f x2 - f x) <= f x - ySublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy, x':Ry_cond:Rabs (y - f x) < Rmin (f x - y) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsHfalse:x1 = x'Hf:Rabs (y - f x) < f x - yFalseSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_neq_x':x1 <> x'x1 <= x' /\ x1 <> x'Sublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'Rabs (x' - x) < epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x' <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy, x':Ry_cond:Rabs (y - f x) < Rmin (f x - y) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsHfalse:x1 = x'Rmin (f x - y) (f x2 - f x) <= f x - ySublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy, x':Ry_cond:Rabs (y - f x) < Rmin (f x - y) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsHfalse:x1 = x'Hf:Rabs (y - f x) < f x - yFalseSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_neq_x':x1 <> x'x1 <= x' /\ x1 <> x'Sublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'Rabs (x' - x) < epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x' <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy, x':Ry_cond:Rabs (y - f x) < Rmin (f x - y) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsHfalse:x1 = x'Hf:Rabs (y - f x) < f x - yFalseSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_neq_x':x1 <> x'x1 <= x' /\ x1 <> x'Sublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'Rabs (x' - x) < epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x' <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy, x':Ry_cond:Rabs (y - f x) < Rmin (f x - y) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsHfalse:x1 = x'Hf:Rabs (y - f x) < f x - yf x - y < f x - ySublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy, x':Ry_cond:Rabs (y - f x) < Rmin (f x - y) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsHfalse:x1 = x'Hf:Rabs (y - f x) < f x - yHfin:f x - y < f x - yFalseSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_neq_x':x1 <> x'x1 <= x' /\ x1 <> x'Sublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'Rabs (x' - x) < epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x' <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy, x':Ry_cond:Rabs (y - f x) < Rmin (f x - y) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsHfalse:x1 = x'Hf:Rabs (y - f x) < f x - yf x - y <= Rabs (y - f x)Sublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy, x':Ry_cond:Rabs (y - f x) < Rmin (f x - y) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsHfalse:x1 = x'Hf:Rabs (y - f x) < f x - yRabs (y - f x) < f x - ySublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy, x':Ry_cond:Rabs (y - f x) < Rmin (f x - y) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsHfalse:x1 = x'Hf:Rabs (y - f x) < f x - yHfin:f x - y < f x - yFalseSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_neq_x':x1 <> x'x1 <= x' /\ x1 <> x'Sublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'Rabs (x' - x) < epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x' <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy, x':Ry_cond:Rabs (y - f x) < Rmin (f x - y) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsHfalse:x1 = x'Hf:Rabs (y - f x) < f x - yf x - y <= Rabs (f x - y)Sublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy, x':Ry_cond:Rabs (y - f x) < Rmin (f x - y) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsHfalse:x1 = x'Hf:Rabs (y - f x) < f x - yRabs (f x - y) = Rabs (y - f x)Sublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy, x':Ry_cond:Rabs (y - f x) < Rmin (f x - y) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsHfalse:x1 = x'Hf:Rabs (y - f x) < f x - yRabs (y - f x) < f x - ySublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy, x':Ry_cond:Rabs (y - f x) < Rmin (f x - y) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsHfalse:x1 = x'Hf:Rabs (y - f x) < f x - yHfin:f x - y < f x - yFalseSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_neq_x':x1 <> x'x1 <= x' /\ x1 <> x'Sublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'Rabs (x' - x) < epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x' <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy, x':Ry_cond:Rabs (y - f x) < Rmin (f x - y) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsHfalse:x1 = x'Hf:Rabs (y - f x) < f x - yRabs (f x - y) = Rabs (y - f x)Sublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy, x':Ry_cond:Rabs (y - f x) < Rmin (f x - y) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsHfalse:x1 = x'Hf:Rabs (y - f x) < f x - yRabs (y - f x) < f x - ySublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy, x':Ry_cond:Rabs (y - f x) < Rmin (f x - y) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsHfalse:x1 = x'Hf:Rabs (y - f x) < f x - yHfin:f x - y < f x - yFalseSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_neq_x':x1 <> x'x1 <= x' /\ x1 <> x'Sublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'Rabs (x' - x) < epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x' <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy, x':Ry_cond:Rabs (y - f x) < Rmin (f x - y) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsHfalse:x1 = x'Hf:Rabs (y - f x) < f x - yRabs (- (f x - y)) = Rabs (y - f x)Sublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy, x':Ry_cond:Rabs (y - f x) < Rmin (f x - y) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsHfalse:x1 = x'Hf:Rabs (y - f x) < f x - yRabs (y - f x) < f x - ySublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy, x':Ry_cond:Rabs (y - f x) < Rmin (f x - y) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsHfalse:x1 = x'Hf:Rabs (y - f x) < f x - yHfin:f x - y < f x - yFalseSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_neq_x':x1 <> x'x1 <= x' /\ x1 <> x'Sublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'Rabs (x' - x) < epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x' <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy, x':Ry_cond:Rabs (y - f x) < Rmin (f x - y) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsHfalse:x1 = x'Hf:Rabs (y - f x) < f x - yRabs (y - f x) < f x - ySublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy, x':Ry_cond:Rabs (y - f x) < Rmin (f x - y) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsHfalse:x1 = x'Hf:Rabs (y - f x) < f x - yHfin:f x - y < f x - yFalseSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_neq_x':x1 <> x'x1 <= x' /\ x1 <> x'Sublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'Rabs (x' - x) < epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x' <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy, x':Ry_cond:Rabs (y - f x) < Rmin (f x - y) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsHfalse:x1 = x'Hf:Rabs (y - f x) < f x - yHfin:f x - y < f x - yFalseSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_neq_x':x1 <> x'x1 <= x' /\ x1 <> x'Sublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'Rabs (x' - x) < epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x' <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_neq_x':x1 <> x'x1 <= x' /\ x1 <> x'Sublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'Rabs (x' - x) < epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x' <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'Rabs (x' - x) < epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x' <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'x - eps < x'Sublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'x'_lb:x - eps < x'Rabs (x' - x) < epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x' <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'x - eps <= x' /\ x - eps <> x'Sublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'x'_lb:x - eps < x'Rabs (x' - x) < epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x' <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'x - eps <= x'Sublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'x - eps <> x'Sublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'x'_lb:x - eps < x'Rabs (x' - x) < epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x' <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'x - eps <> x'Sublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'x'_lb:x - eps < x'Rabs (x' - x) < epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x' <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'x - eps < x'Sublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'x'_lb:x - eps < x'Rabs (x' - x) < epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x' <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'x'_lb:x - eps < x'Rabs (x' - x) < epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x' <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'x'_lb:x - eps < x'x' < x2Sublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'x'_lb:x - eps < x'x'_lt_x2:x' < x2Rabs (x' - x) < epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x' <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'x'_lb:x - eps < x'x' <= x2 /\ x' <> x2Sublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'x'_lb:x - eps < x'x'_lt_x2:x' < x2Rabs (x' - x) < epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x' <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'x'_lb:x - eps < x'x' <> x2Sublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'x'_lb:x - eps < x'x1_neq_x':x' <> x2x' <= x2 /\ x' <> x2Sublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'x'_lb:x - eps < x'x'_lt_x2:x' < x2Rabs (x' - x) < epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x' <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'x'_lb:x - eps < x'Hfalse:x' = x2FalseSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'x'_lb:x - eps < x'x1_neq_x':x' <> x2x' <= x2 /\ x' <> x2Sublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'x'_lb:x - eps < x'x'_lt_x2:x' < x2Rabs (x' - x) < epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x' <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy, x':Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (y - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'x'_lb:x - eps < x'Hfalse:x' = x2FalseSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'x'_lb:x - eps < x'x1_neq_x':x' <> x2x' <= x2 /\ x' <> x2Sublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'x'_lb:x - eps < x'x'_lt_x2:x' < x2Rabs (x' - x) < epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x' <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy, x':Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (y - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'x'_lb:x - eps < x'Hfalse:x' = x2Rabs (y - f x) < y - f xSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy, x':Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (y - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'x'_lb:x - eps < x'Hfalse:x' = x2Hf:Rabs (y - f x) < y - f xFalseSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'x'_lb:x - eps < x'x1_neq_x':x' <> x2x' <= x2 /\ x' <> x2Sublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'x'_lb:x - eps < x'x'_lt_x2:x' < x2Rabs (x' - x) < epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x' <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy, x':Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (y - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'x'_lb:x - eps < x'Hfalse:x' = x2Rabs (y - f x) < Rmin (f x - f x1) (y - f x)Sublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy, x':Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (y - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'x'_lb:x - eps < x'Hfalse:x' = x2Rmin (f x - f x1) (y - f x) <= y - f xSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy, x':Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (y - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'x'_lb:x - eps < x'Hfalse:x' = x2Hf:Rabs (y - f x) < y - f xFalseSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'x'_lb:x - eps < x'x1_neq_x':x' <> x2x' <= x2 /\ x' <> x2Sublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'x'_lb:x - eps < x'x'_lt_x2:x' < x2Rabs (x' - x) < epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x' <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy, x':Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (y - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'x'_lb:x - eps < x'Hfalse:x' = x2Rmin (f x - f x1) (y - f x) <= y - f xSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy, x':Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (y - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'x'_lb:x - eps < x'Hfalse:x' = x2Hf:Rabs (y - f x) < y - f xFalseSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'x'_lb:x - eps < x'x1_neq_x':x' <> x2x' <= x2 /\ x' <> x2Sublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'x'_lb:x - eps < x'x'_lt_x2:x' < x2Rabs (x' - x) < epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x' <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy, x':Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (y - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'x'_lb:x - eps < x'Hfalse:x' = x2Hf:Rabs (y - f x) < y - f xFalseSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'x'_lb:x - eps < x'x1_neq_x':x' <> x2x' <= x2 /\ x' <> x2Sublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'x'_lb:x - eps < x'x'_lt_x2:x' < x2Rabs (x' - x) < epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x' <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy, x':Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (y - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'x'_lb:x - eps < x'Hfalse:x' = x2Hf:Rabs (y - f x) < y - f xy - f x < y - f xSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy, x':Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (y - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'x'_lb:x - eps < x'Hfalse:x' = x2Hf:Rabs (y - f x) < y - f xHfin:y - f x < y - f xFalseSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'x'_lb:x - eps < x'x1_neq_x':x' <> x2x' <= x2 /\ x' <> x2Sublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'x'_lb:x - eps < x'x'_lt_x2:x' < x2Rabs (x' - x) < epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x' <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy, x':Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (y - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'x'_lb:x - eps < x'Hfalse:x' = x2Hf:Rabs (y - f x) < y - f xy - f x <= Rabs (y - f x)Sublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy, x':Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (y - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'x'_lb:x - eps < x'Hfalse:x' = x2Hf:Rabs (y - f x) < y - f xRabs (y - f x) < y - f xSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy, x':Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (y - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'x'_lb:x - eps < x'Hfalse:x' = x2Hf:Rabs (y - f x) < y - f xHfin:y - f x < y - f xFalseSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'x'_lb:x - eps < x'x1_neq_x':x' <> x2x' <= x2 /\ x' <> x2Sublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'x'_lb:x - eps < x'x'_lt_x2:x' < x2Rabs (x' - x) < epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x' <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy, x':Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (y - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'x'_lb:x - eps < x'Hfalse:x' = x2Hf:Rabs (y - f x) < y - f xRabs (y - f x) < y - f xSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy, x':Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (y - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'x'_lb:x - eps < x'Hfalse:x' = x2Hf:Rabs (y - f x) < y - f xHfin:y - f x < y - f xFalseSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'x'_lb:x - eps < x'x1_neq_x':x' <> x2x' <= x2 /\ x' <> x2Sublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'x'_lb:x - eps < x'x'_lt_x2:x' < x2Rabs (x' - x) < epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x' <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy, x':Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (y - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'x'_lb:x - eps < x'Hfalse:x' = x2Hf:Rabs (y - f x) < y - f xHfin:y - f x < y - f xFalseSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'x'_lb:x - eps < x'x1_neq_x':x' <> x2x' <= x2 /\ x' <> x2Sublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'x'_lb:x - eps < x'x'_lt_x2:x' < x2Rabs (x' - x) < epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x' <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'x'_lb:x - eps < x'x1_neq_x':x' <> x2x' <= x2 /\ x' <> x2Sublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'x'_lb:x - eps < x'x'_lt_x2:x' < x2Rabs (x' - x) < epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x' <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'x'_lb:x - eps < x'x'_lt_x2:x' < x2Rabs (x' - x) < epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x' <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'x'_lb:x - eps < x'x'_lt_x2:x' < x2x' < x + epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'x'_lb:x - eps < x'x'_lt_x2:x' < x2x'_ub:x' < x + epsRabs (x' - x) < epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x' <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'x'_lb:x - eps < x'x'_lt_x2:x' < x2x' <= x + eps /\ x' <> x + epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'x'_lb:x - eps < x'x'_lt_x2:x' < x2x'_ub:x' < x + epsRabs (x' - x) < epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x' <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'x'_lb:x - eps < x'x'_lt_x2:x' < x2x' <= x + epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'x'_lb:x - eps < x'x'_lt_x2:x' < x2x' <> x + epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'x'_lb:x - eps < x'x'_lt_x2:x' < x2x'_ub:x' < x + epsRabs (x' - x) < epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x' <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'x'_lb:x - eps < x'x'_lt_x2:x' < x2x' <> x + epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'x'_lb:x - eps < x'x'_lt_x2:x' < x2x'_ub:x' < x + epsRabs (x' - x) < epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x' <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'x'_lb:x - eps < x'x'_lt_x2:x' < x2x' < x + epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'x'_lb:x - eps < x'x'_lt_x2:x' < x2x'_ub:x' < x + epsRabs (x' - x) < epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x' <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx'_encad2:x - eps <= x' <= x + epsx1_lt_x':x1 < x'x'_lb:x - eps < x'x'_lt_x2:x' < x2x'_ub:x' < x + epsRabs (x' - x) < epsSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x' <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x' <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x' <= ubSublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = ylb <= x'Sublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx' <= ubapply Rle_trans with (r2:=x2) ; intuition. Qed.Sublemma:forall x0 y0 z : R, Rmax x0 y0 < z <-> x0 < z /\ y0 < zSublemma2:forall x0 y0 z : R, Rmin x0 y0 > z <-> x0 > z /\ y0 > zSublemma3:forall x0 y0 : R, x0 <= y0 /\ x0 <> y0 -> x0 < y0f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x0 y0 : R, lb <= x0 -> x0 < y0 -> y0 <= ub -> f x0 < f y0f_eq_g:forall x0 : R, lb <= x0 <= ub -> g (f x0) = id x0f_cont_interv:forall a : R, lb <= a <= ub -> continuity_pt f ab:Rb_encad:f lb < b < f ubf_incr_interv2:forall x0 y0 : R, lb <= x0 -> x0 <= y0 -> y0 <= ub -> f x0 <= f y0eps:Reps_pos:eps > 0b_encad_e:f lb <= b <= f ubx:Rx_encad:lb <= x <= ubf_x_b:f x = blb_lt_x:lb < xx_lt_ub:x < ubx1:=Rmax (x - eps) lb:Rx2:=Rmin (x + eps) ub:RHx1:x1 = Rmax (x - eps) lbHx2:x2 = Rmin (x + eps) ubx1_encad:lb <= x1 <= ubx2_encad:lb <= x2 <= ubx_lt_x2:x < x2x1_lt_x:x1 < xy:Ry_cond:Rabs (y - f x) < Rmin (f x - f x1) (f x2 - f x)Main:f x1 <= y <= f x2x1_lt_x2:x1 < x2f_cont_myinterv:forall a : R, x1 <= a <= x2 -> continuity_pt f ax':Rx'_encad:x1 <= x' <= x2f_x'_y:f x' = yx' <= ubforall (f g : R -> R) (lb ub : R), lb < ub -> (forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f y) -> (forall x : R, f lb <= x -> x <= f ub -> comp f g x = id x) -> (forall x : R, f lb <= x -> x <= f ub -> lb <= g x <= ub) -> (forall a : R, lb <= a <= ub -> continuity_pt f a) -> forall b : R, f lb < b < f ub -> continuity_pt g bforall (f g : R -> R) (lb ub : R), lb < ub -> (forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f y) -> (forall x : R, f lb <= x -> x <= f ub -> comp f g x = id x) -> (forall x : R, f lb <= x -> x <= f ub -> lb <= g x <= ub) -> (forall a : R, lb <= a <= ub -> continuity_pt f a) -> forall b : R, f lb < b < f ub -> continuity_pt g bf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f yf_eq_g:forall x : R, f lb <= x -> x <= f ub -> comp f g x = id xg_wf:forall x : R, f lb <= x -> x <= f ub -> lb <= g x <= ub(forall a : R, lb <= a <= ub -> continuity_pt f a) -> forall b : R, f lb < b < f ub -> continuity_pt g bf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f yf_eq_g:forall x : R, f lb <= x -> x <= f ub -> comp f g x = id xg_wf:forall x : R, f lb <= x -> x <= f ub -> lb <= g x <= ubg_eq_f_prelim:(forall x : R, f lb <= x -> x <= f ub -> lb <= g x <= ub) -> forall x : R, lb <= x <= ub -> comp g f x = id x(forall a : R, lb <= a <= ub -> continuity_pt f a) -> forall b : R, f lb < b < f ub -> continuity_pt g bf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f yf_eq_g:forall x : R, f lb <= x -> x <= f ub -> comp f g x = id xg_wf:forall x : R, f lb <= x -> x <= f ub -> lb <= g x <= ubg_eq_f_prelim:(forall x : R, f lb <= x -> x <= f ub -> lb <= g x <= ub) -> forall x : R, lb <= x <= ub -> comp g f x = id xforall x : R, lb <= x <= ub -> comp g f x = id xf, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f yf_eq_g:forall x : R, f lb <= x -> x <= f ub -> comp f g x = id xg_wf:forall x : R, f lb <= x -> x <= f ub -> lb <= g x <= ubg_eq_f_prelim:(forall x : R, f lb <= x -> x <= f ub -> lb <= g x <= ub) -> forall x : R, lb <= x <= ub -> comp g f x = id xg_eq_f:forall x : R, lb <= x <= ub -> comp g f x = id x(forall a : R, lb <= a <= ub -> continuity_pt f a) -> forall b : R, f lb < b < f ub -> continuity_pt g bapply (continuity_pt_recip_prelim f g lb ub lb_lt_ub f_incr_interv g_eq_f). Qed.f, g:R -> Rlb, ub:Rlb_lt_ub:lb < ubf_incr_interv:forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f yf_eq_g:forall x : R, f lb <= x -> x <= f ub -> comp f g x = id xg_wf:forall x : R, f lb <= x -> x <= f ub -> lb <= g x <= ubg_eq_f_prelim:(forall x : R, f lb <= x -> x <= f ub -> lb <= g x <= ub) -> forall x : R, lb <= x <= ub -> comp g f x = id xg_eq_f:forall x : R, lb <= x <= ub -> comp g f x = id x(forall a : R, lb <= a <= ub -> continuity_pt f a) -> forall b : R, f lb < b < f ub -> continuity_pt g b
forall (f g : R -> R) (lb ub x : R) (Prf : forall a : R, g lb <= a <= g ub -> derivable_pt f a), continuity_pt g x -> lb < ub -> lb < x < ub -> forall Prg_incr : g lb <= g x <= g ub, (forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0) -> derive_pt f (g x) (Prf (g x) Prg_incr) <> 0 -> derivable_pt_lim g x (1 / derive_pt f (g x) (Prf (g x) Prg_incr))forall (f g : R -> R) (lb ub x : R) (Prf : forall a : R, g lb <= a <= g ub -> derivable_pt f a), continuity_pt g x -> lb < ub -> lb < x < ub -> forall Prg_incr : g lb <= g x <= g ub, (forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0) -> derive_pt f (g x) (Prf (g x) Prg_incr) <> 0 -> derivable_pt_lim g x (1 / derive_pt f (g x) (Prf (g x) Prg_incr))f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> derivable_pt f ag_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0derivable_pt_lim g x (1 / derive_pt f (g x) (Prf (g x) Prg_incr))f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> derivable_pt f ag_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0lb <= x <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> derivable_pt f ag_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubderivable_pt_lim g x (1 / derive_pt f (g x) (Prf (g x) Prg_incr))f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> derivable_pt f ag_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubderivable_pt_lim g x (1 / derive_pt f (g x) (Prf (g x) Prg_incr))f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> derivable_pt f ag_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) lderivable_pt_lim g x (1 / l)f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> derivable_pt f ag_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) lforall eps : R, 0 < eps -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> derivable_pt f ag_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsexists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> derivable_pt f ag_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:Rexists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> derivable_pt f ag_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> limit1_in (fun x1 : R => / f0 x1) D (/ l0) x0exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> derivable_pt f ag_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> limit1_in (fun x1 : R => / f0 x1) D (/ l0) x0forall eps0 : R, 0 < eps0 -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> derivable_pt f ag_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> limit1_in (fun x1 : R => / f0 x1) D (/ l0) x0Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> derivable_pt f ag_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> limit1_in (fun x1 : R => / f0 x1) D (/ l0) x0eps0:Reps0_pos:0 < eps0exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> derivable_pt f ag_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> limit1_in (fun x1 : R => / f0 x1) D (/ l0) x0Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> derivable_pt f ag_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:forall eps1 : R, 0 < eps1 -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps1eps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> limit1_in (fun x1 : R => / f0 x1) D (/ l0) x0eps0:Reps0_pos:0 < eps0exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> derivable_pt f ag_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> limit1_in (fun x1 : R => / f0 x1) D (/ l0) x0Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> derivable_pt f ag_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:forall eps1 : R, 0 < eps1 -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps1eps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> limit1_in (fun x1 : R => / f0 x1) D (/ l0) x0eps0:Reps0_pos:0 < eps0forall x0 : posreal, (forall h : R, h <> 0 -> Rabs h < x0 -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0) -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> derivable_pt f ag_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> limit1_in (fun x1 : R => / f0 x1) D (/ l0) x0Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> derivable_pt f ag_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:forall eps1 : R, 0 < eps1 -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps1eps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> limit1_in (fun x1 : R => / f0 x1) D (/ l0) x0eps0:Reps0_pos:0 < eps0deltatemp:posrealHtemp:forall h : R, h <> 0 -> Rabs h < deltatemp -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> derivable_pt f ag_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> limit1_in (fun x1 : R => / f0 x1) D (/ l0) x0Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> derivable_pt f ag_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> limit1_in (fun x1 : R => / f0 x1) D (/ l0) x0Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> derivable_pt f ag_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> limit1_in (fun x1 : R => / f0 x1) D (/ l0) x0Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0forall x0 : posreal, (forall h : R, h <> 0 -> Rabs h < x0 -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps) -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> derivable_pt f ag_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> limit1_in (fun x1 : R => / f0 x1) D (/ l0) x0Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0deltatemp:posrealHtemp:forall h : R, h <> 0 -> Rabs h < deltatemp -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsexists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> derivable_pt f ag_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0deltatemp:posrealHtemp:forall h : R, h <> 0 -> Rabs h < deltatemp -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsexists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> derivable_pt f ag_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0deltatemp:posrealHtemp:forall h : R, h <> 0 -> Rabs h < deltatemp -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsHlinv':limit1_in (fun h : R => (f (y + h) - f y) / h) (fun h : R => h <> 0) l 0 -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : Base R_met, (fun h : R => h <> 0) x0 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x0 0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ (fun h : R => (f (y + h) - f y) / h) x0) (/ l) < eps0)exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> derivable_pt f ag_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0deltatemp:posrealHtemp:forall h : R, h <> 0 -> Rabs h < deltatemp -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ R_dist x0 0 < alp -> R_dist ((f (y + x0) - f y) / x0) l < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ R_dist x0 0 < alp -> R_dist (/ ((f (y + x0) - f y) / x0)) (/ l) < eps0)exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> derivable_pt f ag_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0deltatemp:posrealHtemp:forall h : R, h <> 0 -> Rabs h < deltatemp -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> derivable_pt f ag_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0deltatemp:posrealHtemp:forall h : R, h <> 0 -> Rabs h < deltatemp -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> derivable_pt f ag_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0deltatemp:posrealHtemp:forall h : R, h <> 0 -> Rabs h < deltatemp -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h : R => h <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> derivable_pt f ag_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps1 : R, eps1 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps1)Hf_deriv:forall eps1 : R, 0 < eps1 -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps1deltatemp:posrealHtemp:forall h : R, h <> 0 -> Rabs h < deltatemp -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsHlinv':(forall eps1 : R, eps1 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps1)) -> l <> 0 -> forall eps1 : R, eps1 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps1)eps0:Reps0_pos:eps0 > 0exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> derivable_pt f ag_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0deltatemp:posrealHtemp:forall h : R, h <> 0 -> Rabs h < deltatemp -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h : R => h <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> derivable_pt f ag_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps1 : R, eps1 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps1)Hf_deriv:forall eps1 : R, 0 < eps1 -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps1deltatemp:posrealHtemp:forall h : R, h <> 0 -> Rabs h < deltatemp -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsHlinv':(forall eps1 : R, eps1 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps1)) -> l <> 0 -> forall eps1 : R, eps1 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps1)eps0:Reps0_pos:eps0 > 0forall x0 : posreal, (forall h : R, h <> 0 -> Rabs h < x0 -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0) -> exists alp : R, alp > 0 /\ (forall x1 : R, x1 <> 0 /\ Rabs (x1 - 0) < alp -> Rabs ((f (y + x1) - f y) / x1 - l) < eps0)f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> derivable_pt f ag_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0deltatemp:posrealHtemp:forall h : R, h <> 0 -> Rabs h < deltatemp -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h : R => h <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> derivable_pt f ag_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps1 : R, eps1 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps1)Hf_deriv:forall eps1 : R, 0 < eps1 -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps1deltatemp:posrealHtemp:forall h : R, h <> 0 -> Rabs h < deltatemp -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsHlinv':(forall eps1 : R, eps1 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps1)) -> l <> 0 -> forall eps1 : R, eps1 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps1)eps0:Reps0_pos:eps0 > 0deltatemp':posrealHtemp':forall h : R, h <> 0 -> Rabs h < deltatemp' -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> derivable_pt f ag_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0deltatemp:posrealHtemp:forall h : R, h <> 0 -> Rabs h < deltatemp -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h : R => h <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> derivable_pt f ag_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps1 : R, eps1 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps1)Hf_deriv:forall eps1 : R, 0 < eps1 -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps1deltatemp:posrealHtemp:forall h : R, h <> 0 -> Rabs h < deltatemp -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsHlinv':(forall eps1 : R, eps1 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps1)) -> l <> 0 -> forall eps1 : R, eps1 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps1)eps0:Reps0_pos:eps0 > 0deltatemp':posrealHtemp':forall h : R, h <> 0 -> Rabs h < deltatemp' -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0deltatemp' > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < deltatemp' -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> derivable_pt f ag_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0deltatemp:posrealHtemp:forall h : R, h <> 0 -> Rabs h < deltatemp -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h : R => h <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> derivable_pt f ag_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps1 : R, eps1 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps1)Hf_deriv:forall eps1 : R, 0 < eps1 -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps1deltatemp:posrealHtemp:forall h : R, h <> 0 -> Rabs h < deltatemp -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsHlinv':(forall eps1 : R, eps1 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps1)) -> l <> 0 -> forall eps1 : R, eps1 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps1)eps0:Reps0_pos:eps0 > 0deltatemp':posrealHtemp':forall h : R, h <> 0 -> Rabs h < deltatemp' -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0deltatemp' > 0f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> derivable_pt f ag_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps1 : R, eps1 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps1)Hf_deriv:forall eps1 : R, 0 < eps1 -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps1deltatemp:posrealHtemp:forall h : R, h <> 0 -> Rabs h < deltatemp -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsHlinv':(forall eps1 : R, eps1 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps1)) -> l <> 0 -> forall eps1 : R, eps1 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps1)eps0:Reps0_pos:eps0 > 0deltatemp':posrealHtemp':forall h : R, h <> 0 -> Rabs h < deltatemp' -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < deltatemp' -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> derivable_pt f ag_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0deltatemp:posrealHtemp:forall h : R, h <> 0 -> Rabs h < deltatemp -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h : R => h <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> derivable_pt f ag_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps1 : R, eps1 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps1)Hf_deriv:forall eps1 : R, 0 < eps1 -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps1deltatemp:posrealHtemp:forall h : R, h <> 0 -> Rabs h < deltatemp -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsHlinv':(forall eps1 : R, eps1 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps1)) -> l <> 0 -> forall eps1 : R, eps1 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps1)eps0:Reps0_pos:eps0 > 0deltatemp':posrealHtemp':forall h : R, h <> 0 -> Rabs h < deltatemp' -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < deltatemp' -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> derivable_pt f ag_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0deltatemp:posrealHtemp:forall h : R, h <> 0 -> Rabs h < deltatemp -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h : R => h <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> derivable_pt f ag_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps1 : R, eps1 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps1)Hf_deriv:forall eps1 : R, 0 < eps1 -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps1deltatemp:posrealHtemp:forall h : R, h <> 0 -> Rabs h < deltatemp -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsHlinv':(forall eps1 : R, eps1 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps1)) -> l <> 0 -> forall eps1 : R, eps1 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps1)eps0:Reps0_pos:eps0 > 0deltatemp':posrealHtemp':forall h : R, h <> 0 -> Rabs h < deltatemp' -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0htemp:Rcond:htemp <> 0 /\ Rabs (htemp - 0) < deltatemp'Rabs ((f (y + htemp) - f y) / htemp - l) < eps0f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> derivable_pt f ag_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0deltatemp:posrealHtemp:forall h : R, h <> 0 -> Rabs h < deltatemp -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h : R => h <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> derivable_pt f ag_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps1 : R, eps1 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps1)Hf_deriv:forall eps1 : R, 0 < eps1 -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps1deltatemp:posrealHtemp:forall h : R, h <> 0 -> Rabs h < deltatemp -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsHlinv':(forall eps1 : R, eps1 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps1)) -> l <> 0 -> forall eps1 : R, eps1 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps1)eps0:Reps0_pos:eps0 > 0deltatemp':posrealHtemp':forall h : R, h <> 0 -> Rabs h < deltatemp' -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0htemp:Rcond:htemp <> 0 /\ Rabs (htemp - 0) < deltatemp'htemp <> 0f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> derivable_pt f ag_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps1 : R, eps1 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps1)Hf_deriv:forall eps1 : R, 0 < eps1 -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps1deltatemp:posrealHtemp:forall h : R, h <> 0 -> Rabs h < deltatemp -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsHlinv':(forall eps1 : R, eps1 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps1)) -> l <> 0 -> forall eps1 : R, eps1 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps1)eps0:Reps0_pos:eps0 > 0deltatemp':posrealHtemp':forall h : R, h <> 0 -> Rabs h < deltatemp' -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0htemp:Rcond:htemp <> 0 /\ Rabs (htemp - 0) < deltatemp'Rabs htemp < deltatemp'f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> derivable_pt f ag_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0deltatemp:posrealHtemp:forall h : R, h <> 0 -> Rabs h < deltatemp -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h : R => h <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> derivable_pt f ag_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps1 : R, eps1 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps1)Hf_deriv:forall eps1 : R, 0 < eps1 -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps1deltatemp:posrealHtemp:forall h : R, h <> 0 -> Rabs h < deltatemp -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsHlinv':(forall eps1 : R, eps1 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps1)) -> l <> 0 -> forall eps1 : R, eps1 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps1)eps0:Reps0_pos:eps0 > 0deltatemp':posrealHtemp':forall h : R, h <> 0 -> Rabs h < deltatemp' -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0htemp:Rcond:htemp <> 0 /\ Rabs (htemp - 0) < deltatemp'Rabs htemp < deltatemp'f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> derivable_pt f ag_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0deltatemp:posrealHtemp:forall h : R, h <> 0 -> Rabs h < deltatemp -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h : R => h <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> derivable_pt f ag_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps1 : R, eps1 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps1)Hf_deriv:forall eps1 : R, 0 < eps1 -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps1deltatemp:posrealHtemp:forall h : R, h <> 0 -> Rabs h < deltatemp -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsHlinv':(forall eps1 : R, eps1 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps1)) -> l <> 0 -> forall eps1 : R, eps1 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps1)eps0:Reps0_pos:eps0 > 0deltatemp':posrealHtemp':forall h : R, h <> 0 -> Rabs h < deltatemp' -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0htemp:Rcond:htemp <> 0 /\ Rabs (htemp - 0) < deltatemp'Rabs (htemp - 0) < deltatemp'f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> derivable_pt f ag_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps1 : R, eps1 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps1)Hf_deriv:forall eps1 : R, 0 < eps1 -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps1deltatemp:posrealHtemp:forall h : R, h <> 0 -> Rabs h < deltatemp -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsHlinv':(forall eps1 : R, eps1 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps1)) -> l <> 0 -> forall eps1 : R, eps1 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps1)eps0:Reps0_pos:eps0 > 0deltatemp':posrealHtemp':forall h : R, h <> 0 -> Rabs h < deltatemp' -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0htemp:Rcond:htemp <> 0 /\ Rabs (htemp - 0) < deltatemp'htemp - 0 = htempf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> derivable_pt f ag_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0deltatemp:posrealHtemp:forall h : R, h <> 0 -> Rabs h < deltatemp -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h : R => h <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> derivable_pt f ag_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps1 : R, eps1 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps1)Hf_deriv:forall eps1 : R, 0 < eps1 -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps1deltatemp:posrealHtemp:forall h : R, h <> 0 -> Rabs h < deltatemp -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsHlinv':(forall eps1 : R, eps1 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps1)) -> l <> 0 -> forall eps1 : R, eps1 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps1)eps0:Reps0_pos:eps0 > 0deltatemp':posrealHtemp':forall h : R, h <> 0 -> Rabs h < deltatemp' -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0htemp:Rcond:htemp <> 0 /\ Rabs (htemp - 0) < deltatemp'htemp - 0 = htempf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> derivable_pt f ag_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0deltatemp:posrealHtemp:forall h : R, h <> 0 -> Rabs h < deltatemp -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h : R => h <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> derivable_pt f ag_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0deltatemp:posrealHtemp:forall h : R, h <> 0 -> Rabs h < deltatemp -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h : R => h <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> derivable_pt f ag_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0deltatemp:posrealHtemp:forall h : R, h <> 0 -> Rabs h < deltatemp -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h : R => h <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)l <> 0f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> derivable_pt f ag_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0deltatemp:posrealHtemp:forall h : R, h <> 0 -> Rabs h < deltatemp -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h : R => h <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> derivable_pt f ag_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0deltatemp:posrealHtemp:forall h : R, h <> 0 -> Rabs h < deltatemp -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h : R => h <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)l_null:l = 0Falsef, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> derivable_pt f ag_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0deltatemp:posrealHtemp:forall h : R, h <> 0 -> Rabs h < deltatemp -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h : R => h <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> derivable_pt f ag_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) 0eps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0deltatemp:posrealHtemp:forall h : R, h <> 0 -> Rabs h < deltatemp -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h : R => h <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)l_null:l = 0Falsef, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> derivable_pt f ag_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0deltatemp:posrealHtemp:forall h : R, h <> 0 -> Rabs h < deltatemp -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h : R => h <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> derivable_pt f ag_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) 0eps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0deltatemp:posrealHtemp:forall h : R, h <> 0 -> Rabs h < deltatemp -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h : R => h <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)l_null:l = 0derive_pt f (g x) (Prf (g x) Prg_incr) = 0f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> derivable_pt f ag_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0deltatemp:posrealHtemp:forall h : R, h <> 0 -> Rabs h < deltatemp -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h : R => h <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> derivable_pt f ag_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) 0eps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0deltatemp:posrealHtemp:forall h : R, h <> 0 -> Rabs h < deltatemp -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h : R => h <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)l_null:l = 0derivable_pt_lim f (g x) 0f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> derivable_pt f ag_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0deltatemp:posrealHtemp:forall h : R, h <> 0 -> Rabs h < deltatemp -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h : R => h <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> derivable_pt f ag_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0deltatemp:posrealHtemp:forall h : R, h <> 0 -> Rabs h < deltatemp -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h : R => h <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> derivable_pt f ag_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0deltatemp:posrealHtemp:forall h : R, h <> 0 -> Rabs h < deltatemp -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h : R => h <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0forall x0 : R, x0 > 0 /\ (forall x1 : R, x1 <> 0 /\ Rabs (x1 - 0) < x0 -> Rabs (/ ((f (y + x1) - f y) / x1) - / l) < eps) -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> derivable_pt f ag_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0deltatemp:posrealHtemp:forall h : R, h <> 0 -> Rabs h < deltatemp -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h : R => h <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Rcond:alpha > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps)exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> derivable_pt f ag_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0deltatemp:posrealHtemp:forall h : R, h <> 0 -> Rabs h < deltatemp -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h : R => h <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsexists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (a + h) - f a) / h - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0deltatemp:posrealHtemp:forall h : R, h <> 0 -> Rabs h < deltatemp -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h : R => h <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsexists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (a + h) - f a) / h - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0deltatemp:posrealHtemp:forall h : R, h <> 0 -> Rabs h < deltatemp -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h : R => h <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsforall x0 : posreal, (forall h : R, h <> 0 -> Rabs h < x0 -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps) -> exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((f (a + h) - f a) / h - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0deltatemp:posrealHtemp:forall h : R, h <> 0 -> Rabs h < deltatemp -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h : R => h <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsexists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((f (a + h) - f a) / h - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0deltatemp:posrealHtemp:forall h : R, h <> 0 -> Rabs h < deltatemp -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h : R => h <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsg_cont:continuity_pt g xexists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((f (a + h) - f a) / h - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0deltatemp:posrealHtemp:forall h : R, h <> 0 -> Rabs h < deltatemp -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h : R => h <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsg_cont:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < alp -> dist R_met (g x0) (g x) < eps0)exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((f (a + h) - f a) / h - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0deltatemp:posrealHtemp:forall h : R, h <> 0 -> Rabs h < deltatemp -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h : R => h <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsg_cont:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < alp -> dist R_met (g x0) (g x) < eps0)mydelta:=Rmin delta alpha:Rexists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((f (a + h) - f a) / h - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0deltatemp:posrealHtemp:forall h : R, h <> 0 -> Rabs h < deltatemp -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h : R => h <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsg_cont:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < alp -> dist R_met (g x0) (g x) < eps0)mydelta:=Rmin delta alpha:Rmydelta > 0f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((f (a + h) - f a) / h - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0deltatemp:posrealHtemp:forall h : R, h <> 0 -> Rabs h < deltatemp -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h : R => h <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsg_cont:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < alp -> dist R_met (g x0) (g x) < eps0)mydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((f (a + h) - f a) / h - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0deltatemp:posrealHtemp:forall h : R, h <> 0 -> Rabs h < deltatemp -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h : R => h <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsg_cont:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < alp -> dist R_met (g x0) (g x) < eps0)mydelta:=Rmin delta alpha:R(if Rle_dec delta alpha then delta else alpha) > 0f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((f (a + h) - f a) / h - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0deltatemp:posrealHtemp:forall h : R, h <> 0 -> Rabs h < deltatemp -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h : R => h <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsg_cont:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < alp -> dist R_met (g x0) (g x) < eps0)mydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((f (a + h) - f a) / h - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0deltatemp:posrealHtemp:forall h : R, h <> 0 -> Rabs h < deltatemp -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h : R => h <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsg_cont:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < alp -> dist R_met (g x0) (g x) < eps0)mydelta:=Rmin delta alpha:Rdelta <= alpha -> delta > 0f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((f (a + h) - f a) / h - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0deltatemp:posrealHtemp:forall h : R, h <> 0 -> Rabs h < deltatemp -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h : R => h <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsg_cont:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < alp -> dist R_met (g x0) (g x) < eps0)mydelta:=Rmin delta alpha:R~ delta <= alpha -> alpha > 0f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((f (a + h) - f a) / h - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0deltatemp:posrealHtemp:forall h : R, h <> 0 -> Rabs h < deltatemp -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h : R => h <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsg_cont:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < alp -> dist R_met (g x0) (g x) < eps0)mydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((f (a + h) - f a) / h - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0deltatemp:posrealHtemp:forall h : R, h <> 0 -> Rabs h < deltatemp -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h : R => h <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsg_cont:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < alp -> dist R_met (g x0) (g x) < eps0)mydelta:=Rmin delta alpha:R~ delta <= alpha -> alpha > 0f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((f (a + h) - f a) / h - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0deltatemp:posrealHtemp:forall h : R, h <> 0 -> Rabs h < deltatemp -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h : R => h <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsg_cont:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < alp -> dist R_met (g x0) (g x) < eps0)mydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((f (a + h) - f a) / h - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0deltatemp:posrealHtemp:forall h : R, h <> 0 -> Rabs h < deltatemp -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h : R => h <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsg_cont:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < alp -> dist R_met (g x0) (g x) < eps0)mydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((f (a + h) - f a) / h - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0deltatemp:posrealHtemp:forall h : R, h <> 0 -> Rabs h < deltatemp -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h : R => h <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsg_cont:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < alp -> dist R_met (g x0) (g x) < eps0)mydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0forall x0 : R, x0 > 0 /\ (forall x1 : Base R_met, D_x no_cond x x1 /\ dist R_met x1 x < x0 -> dist R_met (g x1) (g x) < mydelta) -> exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((f (a + h) - f a) / h - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0deltatemp:posrealHtemp:forall h : R, h <> 0 -> Rabs h < deltatemp -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h : R => h <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsg_cont:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < alp -> dist R_met (g x0) (g x) < eps0)mydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rnew_g_cont:delta' > 0 /\ (forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydelta)exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((f (a + h) - f a) / h - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0deltatemp:posrealHtemp:forall h : R, h <> 0 -> Rabs h < deltatemp -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h : R => h <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsg_cont:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < alp -> dist R_met (g x0) (g x) < eps0)mydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rnew_g_cont:delta' > 0 /\ (forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydelta)delta'_pos:delta' > 0exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((f (a + h) - f a) / h - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0deltatemp:posrealHtemp:forall h : R, h <> 0 -> Rabs h < deltatemp -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h : R => h <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltaexists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((f (a + h) - f a) / h - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0deltatemp:posrealHtemp:forall h : R, h <> 0 -> Rabs h < deltatemp -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h : R => h <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rexists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((f (a + h) - f a) / h - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0deltatemp:posrealHtemp:forall h : R, h <> 0 -> Rabs h < deltatemp -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h : R => h <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta'' > 0f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((f (a + h) - f a) / h - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0deltatemp:posrealHtemp:forall h : R, h <> 0 -> Rabs h < deltatemp -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h : R => h <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((f (a + h) - f a) / h - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0deltatemp:posrealHtemp:forall h : R, h <> 0 -> Rabs h < deltatemp -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h : R => h <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):RRmin delta' (Rmin (x - lb) (ub - x)) > 0f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((f (a + h) - f a) / h - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0deltatemp:posrealHtemp:forall h : R, h <> 0 -> Rabs h < deltatemp -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h : R => h <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((f (a + h) - f a) / h - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0deltatemp:posrealHtemp:forall h : R, h <> 0 -> Rabs h < deltatemp -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h : R => h <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((f (a + h) - f a) / h - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0deltatemp:posrealHtemp:forall h : R, h <> 0 -> Rabs h < deltatemp -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h : R => h <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealexists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((f (a + h) - f a) / h - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps0deltatemp:posrealHtemp:forall h : R, h <> 0 -> Rabs h < deltatemp -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h : R => h <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealforall h : R, h <> 0 -> Rabs h < delta'' -> Rabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Rabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''lb <= x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubRabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0lb <= x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubRabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m0 return (Base m0 -> Base m0 -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m0 return (Base m0 -> Base m0 -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''m, n:RHyp_abs:Rabs m < Rabs ny_pos:n > 0m < nf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0lb <= x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubRabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m0 return (Base m0 -> Base m0 -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m0 return (Base m0 -> Base m0 -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''m, n:RHyp_abs:Rabs m < Rabs ny_pos:n > 0m < Rabs nf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m0 return (Base m0 -> Base m0 -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m0 return (Base m0 -> Base m0 -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''m, n:RHyp_abs:Rabs m < Rabs ny_pos:n > 0Rabs n <= nf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0lb <= x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubRabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m0 return (Base m0 -> Base m0 -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m0 return (Base m0 -> Base m0 -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''m, n:RHyp_abs:Rabs m < Rabs ny_pos:n > 0Rabs n <= nf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0lb <= x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubRabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0lb <= x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubRabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0lb <= x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ublb <= x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubRabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0lb <= x + hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ublb <= x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubRabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0forall x0 y0 z : R, - z <= y0 - x0 -> x0 <= y0 + zf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0Sublemma:forall x0 y0 z : R, - z <= y0 - x0 -> x0 <= y0 + zlb <= x + hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ublb <= x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubRabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0Sublemma:forall x0 y0 z : R, - z <= y0 - x0 -> x0 <= y0 + zlb <= x + hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ublb <= x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubRabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0Sublemma:forall x0 y0 z : R, - z <= y0 - x0 -> x0 <= y0 + z- h <= x - lbf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ublb <= x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubRabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0Sublemma:forall x0 y0 z : R, - z <= y0 - x0 -> x0 <= y0 + zRabs (- h) < Rabs (x - lb)f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0Sublemma:forall x0 y0 z : R, - z <= y0 - x0 -> x0 <= y0 + zx - lb > 0f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ublb <= x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubRabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0Sublemma:forall x0 y0 z : R, - z <= y0 - x0 -> x0 <= y0 + zRabs h < Rabs (x - lb)f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0Sublemma:forall x0 y0 z : R, - z <= y0 - x0 -> x0 <= y0 + zx - lb > 0f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ublb <= x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubRabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0Sublemma:forall x0 y0 z : R, - z <= y0 - x0 -> x0 <= y0 + zRabs h < Rmin delta' (Rmin (x - lb) (ub - x))f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0Sublemma:forall x0 y0 z : R, - z <= y0 - x0 -> x0 <= y0 + zRmin delta' (Rmin (x - lb) (ub - x)) <= Rmin (x - lb) (ub - x)f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0Sublemma:forall x0 y0 z : R, - z <= y0 - x0 -> x0 <= y0 + zx - lb > 0f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ublb <= x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubRabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0Sublemma:forall x0 y0 z : R, - z <= y0 - x0 -> x0 <= y0 + zRabs h < delta''f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0Sublemma:forall x0 y0 z : R, - z <= y0 - x0 -> x0 <= y0 + zdelta'' <= Rmin delta' (Rmin (x - lb) (ub - x))f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0Sublemma:forall x0 y0 z : R, - z <= y0 - x0 -> x0 <= y0 + zRmin delta' (Rmin (x - lb) (ub - x)) <= Rmin (x - lb) (ub - x)f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0Sublemma:forall x0 y0 z : R, - z <= y0 - x0 -> x0 <= y0 + zx - lb > 0f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ublb <= x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubRabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0Sublemma:forall x0 y0 z : R, - z <= y0 - x0 -> x0 <= y0 + zdelta'' <= Rmin delta' (Rmin (x - lb) (ub - x))f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0Sublemma:forall x0 y0 z : R, - z <= y0 - x0 -> x0 <= y0 + zRmin delta' (Rmin (x - lb) (ub - x)) <= Rmin (x - lb) (ub - x)f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0Sublemma:forall x0 y0 z : R, - z <= y0 - x0 -> x0 <= y0 + zx - lb > 0f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ublb <= x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubRabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0Sublemma:forall x0 y0 z : R, - z <= y0 - x0 -> x0 <= y0 + zRmin delta' (Rmin (x - lb) (ub - x)) <= Rmin (x - lb) (ub - x)f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0Sublemma:forall x0 y0 z : R, - z <= y0 - x0 -> x0 <= y0 + zx - lb > 0f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ublb <= x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubRabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0Sublemma:forall x0 y0 z : R, - z <= y0 - x0 -> x0 <= y0 + zx - lb > 0f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ublb <= x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubRabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0Sublemma:forall x0 y0 z : R, - z <= y0 - x0 -> x0 <= y0 + zx > lbf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ublb <= x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubRabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ublb <= x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubRabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0forall x0 y0 z : R, y0 <= z - x0 -> x0 + y0 <= zf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0Sublemma:forall x0 y0 z : R, y0 <= z - x0 -> x0 + y0 <= zx + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ublb <= x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubRabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0Sublemma:forall x0 y0 z : R, y0 <= z - x0 -> x0 + y0 <= zx + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ublb <= x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubRabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0Sublemma:forall x0 y0 z : R, y0 <= z - x0 -> x0 + y0 <= zh <= ub - xf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ublb <= x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubRabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0Sublemma:forall x0 y0 z : R, y0 <= z - x0 -> x0 + y0 <= zRabs h < Rabs (ub - x)f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0Sublemma:forall x0 y0 z : R, y0 <= z - x0 -> x0 + y0 <= zub - x > 0f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ublb <= x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubRabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0Sublemma:forall x0 y0 z : R, y0 <= z - x0 -> x0 + y0 <= zub - x > 0f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ublb <= x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubRabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0Sublemma:forall x0 y0 z : R, y0 <= z - x0 -> x0 + y0 <= z0 < delta''f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0Sublemma:forall x0 y0 z : R, y0 <= z - x0 -> x0 + y0 <= zdelta'' <= ub - xf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ublb <= x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubRabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0Sublemma:forall x0 y0 z : R, y0 <= z - x0 -> x0 + y0 <= zdelta'' <= ub - xf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ublb <= x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubRabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0Sublemma:forall x0 y0 z : R, y0 <= z - x0 -> x0 + y0 <= zdelta'' <= Rmin delta' (Rmin (x - lb) (ub - x))f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0Sublemma:forall x0 y0 z : R, y0 <= z - x0 -> x0 + y0 <= zRmin delta' (Rmin (x - lb) (ub - x)) <= ub - xf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ublb <= x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubRabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0Sublemma:forall x0 y0 z : R, y0 <= z - x0 -> x0 + y0 <= zRmin delta' (Rmin (x - lb) (ub - x)) <= ub - xf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ublb <= x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubRabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0Sublemma:forall x0 y0 z : R, y0 <= z - x0 -> x0 + y0 <= zRmin delta' (Rmin (x - lb) (ub - x)) <= Rmin (x - lb) (ub - x)f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0Sublemma:forall x0 y0 z : R, y0 <= z - x0 -> x0 + y0 <= zRmin (x - lb) (ub - x) <= ub - xf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ublb <= x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubRabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0Sublemma:forall x0 y0 z : R, y0 <= z - x0 -> x0 + y0 <= zRmin (x - lb) (ub - x) <= ub - xf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ublb <= x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubRabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ublb <= x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubRabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ublb <= x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ub1 / (h / (g (x + h) - g x)) = (g (x + h) - g x) / hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubRabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ubh = comp f g (x + h) - comp f g xf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xlb <= x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ub1 / (h / (g (x + h) - g x)) = (g (x + h) - g x) / hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubRabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ubh = id (x + h) - comp f g xf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ublb <= x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xlb <= x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ub1 / (h / (g (x + h) - g x)) = (g (x + h) - g x) / hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubRabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ubh = x + h - xf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ublb <= x <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ublb <= x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xlb <= x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ub1 / (h / (g (x + h) - g x)) = (g (x + h) - g x) / hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubRabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ubh = h + 0f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ublb <= x <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ublb <= x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xlb <= x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ub1 / (h / (g (x + h) - g x)) = (g (x + h) - g x) / hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubRabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ublb <= x <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ublb <= x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xlb <= x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ub1 / (h / (g (x + h) - g x)) = (g (x + h) - g x) / hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubRabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ublb <= x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xlb <= x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ub1 / (h / (g (x + h) - g x)) = (g (x + h) - g x) / hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubRabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xlb <= x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ub1 / (h / (g (x + h) - g x)) = (g (x + h) - g x) / hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubRabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xlb <= x + hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ub1 / (h / (g (x + h) - g x)) = (g (x + h) - g x) / hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubRabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xforall x0 y0 z : R, - z <= y0 - x0 -> x0 <= y0 + zf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xSublemma:forall x0 y0 z : R, - z <= y0 - x0 -> x0 <= y0 + zlb <= x + hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ub1 / (h / (g (x + h) - g x)) = (g (x + h) - g x) / hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubRabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xSublemma:forall x0 y0 z : R, - z <= y0 - x0 -> x0 <= y0 + zlb <= x + hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ub1 / (h / (g (x + h) - g x)) = (g (x + h) - g x) / hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubRabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xSublemma:forall x0 y0 z : R, - z <= y0 - x0 -> x0 <= y0 + zRabs (- h) < Rabs (x - lb)f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xSublemma:forall x0 y0 z : R, - z <= y0 - x0 -> x0 <= y0 + zx - lb > 0f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ub1 / (h / (g (x + h) - g x)) = (g (x + h) - g x) / hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubRabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xSublemma:forall x0 y0 z : R, - z <= y0 - x0 -> x0 <= y0 + zRabs h < Rabs (x - lb)f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xSublemma:forall x0 y0 z : R, - z <= y0 - x0 -> x0 <= y0 + zx - lb > 0f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ub1 / (h / (g (x + h) - g x)) = (g (x + h) - g x) / hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubRabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xSublemma:forall x0 y0 z : R, - z <= y0 - x0 -> x0 <= y0 + zx - lb > 0f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ub1 / (h / (g (x + h) - g x)) = (g (x + h) - g x) / hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubRabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xSublemma:forall x0 y0 z : R, - z <= y0 - x0 -> x0 <= y0 + zx > lbf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ub1 / (h / (g (x + h) - g x)) = (g (x + h) - g x) / hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubRabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ub1 / (h / (g (x + h) - g x)) = (g (x + h) - g x) / hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubRabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ubh <> 0 /\ g (x + h) - g x <> 0f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubRabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ubh <> 0f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ubg (x + h) - g x <> 0f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubRabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ubg (x + h) - g x <> 0f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubRabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ubHfalse:g (x + h) - g x = 0Falsef, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubRabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ubHfalse:g (x + h) - g x = 0Hf:g (x + h) = g xFalsef, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubRabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ubHfalse:g (x + h) - g x = 0Hf:g (x + h) = g xcomp f g (x + h) = comp f g xf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ubHfalse:g (x + h) - g x = 0Hf:g (x + h) = g xH0:comp f g (x + h) = comp f g xFalsef, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubRabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ubHfalse:g (x + h) - g x = 0Hf:g (x + h) = g xH0:comp f g (x + h) = comp f g xFalsef, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubRabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ubHfalse:g (x + h) - g x = 0Hf:g (x + h) = g xH0:comp f g (x + h) = comp f g xx + h = xf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ubHfalse:g (x + h) - g x = 0Hf:g (x + h) = g xH0:comp f g (x + h) = comp f g xMain:x + h = xFalsef, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubRabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ubHfalse:g (x + h) - g x = 0Hf:g (x + h) = g xH0:comp f g (x + h) = comp f g xid (x + h) = xf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ubHfalse:g (x + h) - g x = 0Hf:g (x + h) = g xH0:comp f g (x + h) = comp f g xMain:x + h = xFalsef, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubRabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ubHfalse:g (x + h) - g x = 0Hf:g (x + h) = g xH0:comp f g (x + h) = comp f g xid (x + h) = id xf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ubHfalse:g (x + h) - g x = 0Hf:g (x + h) = g xH0:comp f g (x + h) = comp f g xMain:x + h = xFalsef, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubRabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ubHfalse:g (x + h) - g x = 0Hf:g (x + h) = g xH0:comp f g (x + h) = comp f g xcomp f g (x + h) = id xf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ubHfalse:g (x + h) - g x = 0Hf:g (x + h) = g xH0:comp f g (x + h) = comp f g xlb <= x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ubHfalse:g (x + h) - g x = 0Hf:g (x + h) = g xH0:comp f g (x + h) = comp f g xMain:x + h = xFalsef, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubRabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ubHfalse:g (x + h) - g x = 0Hf:g (x + h) = g xH0:comp f g (x + h) = comp f g xcomp f g (x + h) = comp f g xf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ubHfalse:g (x + h) - g x = 0Hf:g (x + h) = g xH0:comp f g (x + h) = comp f g xlb <= x <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ubHfalse:g (x + h) - g x = 0Hf:g (x + h) = g xH0:comp f g (x + h) = comp f g xlb <= x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ubHfalse:g (x + h) - g x = 0Hf:g (x + h) = g xH0:comp f g (x + h) = comp f g xMain:x + h = xFalsef, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubRabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ubHfalse:g (x + h) - g x = 0Hf:g (x + h) = g xH0:comp f g (x + h) = comp f g xlb <= x <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ubHfalse:g (x + h) - g x = 0Hf:g (x + h) = g xH0:comp f g (x + h) = comp f g xlb <= x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ubHfalse:g (x + h) - g x = 0Hf:g (x + h) = g xH0:comp f g (x + h) = comp f g xMain:x + h = xFalsef, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubRabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ubHfalse:g (x + h) - g x = 0Hf:g (x + h) = g xH0:comp f g (x + h) = comp f g xlb <= x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ubHfalse:g (x + h) - g x = 0Hf:g (x + h) = g xH0:comp f g (x + h) = comp f g xMain:x + h = xFalsef, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubRabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ubHfalse:g (x + h) - g x = 0Hf:g (x + h) = g xH0:comp f g (x + h) = comp f g xMain:x + h = xFalsef, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubRabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ubHfalse:g (x + h) - g x = 0Hf:g (x + h) = g xH0:comp f g (x + h) = comp f g xMain:x + h = xh = 0f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ubHfalse:g (x + h) - g x = 0Hf:g (x + h) = g xH0:comp f g (x + h) = comp f g xMain:x + h = xH1:h = 0Falsef, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubRabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''Sublemma2:forall x0 y0 : R, Rabs x0 < Rabs y0 -> y0 > 0 -> x0 < y0H:lb <= x + h <= ubHfalse:g (x + h) - g x = 0Hf:g (x + h) = g xH0:comp f g (x + h) = comp f g xMain:x + h = xH1:h = 0Falsef, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubRabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubRabs ((g (x + h) - g x) / h - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubRabs (1 / (h / (g (x + h) - g x)) - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ub1 / (h / (g (x + h) - g x)) = (g (x + h) - g x) / hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubh = comp f g (x + h) - comp f g xf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xRabs (1 / (h / (g (x + h) - g x)) - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ub1 / (h / (g (x + h) - g x)) = (g (x + h) - g x) / hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubh = id (x + h) - comp f g xf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ublb <= x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xRabs (1 / (h / (g (x + h) - g x)) - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ub1 / (h / (g (x + h) - g x)) = (g (x + h) - g x) / hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubh = id (x + h) - id xf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ublb <= x <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ublb <= x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xRabs (1 / (h / (g (x + h) - g x)) - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ub1 / (h / (g (x + h) - g x)) = (g (x + h) - g x) / hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ublb <= x <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ublb <= x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xRabs (1 / (h / (g (x + h) - g x)) - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ub1 / (h / (g (x + h) - g x)) = (g (x + h) - g x) / hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ublb <= x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xRabs (1 / (h / (g (x + h) - g x)) - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ub1 / (h / (g (x + h) - g x)) = (g (x + h) - g x) / hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xRabs (1 / (h / (g (x + h) - g x)) - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ub1 / (h / (g (x + h) - g x)) = (g (x + h) - g x) / hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xRabs (1 / ((comp f g (x + h) - comp f g x) / (g (x + h) - g x)) - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ub1 / (h / (g (x + h) - g x)) = (g (x + h) - g x) / hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xRabs (1 / ((f (g (x + h)) - f (g x)) / (g (x + h) - g x)) - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ub1 / (h / (g (x + h) - g x)) = (g (x + h) - g x) / hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xRabs (1 / ((f (g x + (g (x + h) - g x)) - f (g x)) / (g x + (g (x + h) - g x) - g x)) - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ub1 / (h / (g (x + h) - g x)) = (g (x + h) - g x) / hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:RRabs (1 / ((f (g x + (g (x + h) - g x)) - f (g x)) / (g x + (g (x + h) - g x) - g x)) - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ub1 / (h / (g (x + h) - g x)) = (g (x + h) - g x) / hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:RRabs (1 / ((f (g x + h') - f (g x)) / (g x + h' - g x)) - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ub1 / (h / (g (x + h) - g x)) = (g (x + h) - g x) / hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:RRabs (1 / ((f (g x + h') - f (g x)) / h') - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ub1 / (h / (g (x + h) - g x)) = (g (x + h) - g x) / hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rh' <> 0f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rh'_neq:h' <> 0Rabs (1 / ((f (g x + h') - f (g x)) / h') - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ub1 / (h / (g (x + h) - g x)) = (g (x + h) - g x) / hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rg (x + h) - g x <> 0f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rh'_neq:h' <> 0Rabs (1 / ((f (g x + h') - f (g x)) / h') - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ub1 / (h / (g (x + h) - g x)) = (g (x + h) - g x) / hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:RHfalse:g (x + h) - g x = 0Falsef, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rh'_neq:h' <> 0Rabs (1 / ((f (g x + h') - f (g x)) / h') - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ub1 / (h / (g (x + h) - g x)) = (g (x + h) - g x) / hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:RHfalse:g (x + h) = g xFalsef, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rh'_neq:h' <> 0Rabs (1 / ((f (g x + h') - f (g x)) / h') - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ub1 / (h / (g (x + h) - g x)) = (g (x + h) - g x) / hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:RHfalse:g (x + h) = g xcomp f g (x + h) = comp f g xf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:RHfalse:g (x + h) = g xHfalse':comp f g (x + h) = comp f g xFalsef, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rh'_neq:h' <> 0Rabs (1 / ((f (g x + h') - f (g x)) / h') - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ub1 / (h / (g (x + h) - g x)) = (g (x + h) - g x) / hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:RHfalse:g (x + h) = g xHfalse':comp f g (x + h) = comp f g xFalsef, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rh'_neq:h' <> 0Rabs (1 / ((f (g x + h') - f (g x)) / h') - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ub1 / (h / (g (x + h) - g x)) = (g (x + h) - g x) / hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:RHfalse:g (x + h) = g xHfalse':id (x + h) = id xFalsef, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:RHfalse:g (x + h) = g xHfalse':id (x + h) = comp f g xlb <= x <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:RHfalse:g (x + h) = g xHfalse':id (x + h) = comp f g xlb <= x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:RHfalse:g (x + h) = g xHfalse':comp f g (x + h) = comp f g xlb <= x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rh'_neq:h' <> 0Rabs (1 / ((f (g x + h') - f (g x)) / h') - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ub1 / (h / (g (x + h) - g x)) = (g (x + h) - g x) / hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:RHfalse:g (x + h) = g xHfalse':x + h = xFalsef, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:RHfalse:g (x + h) = g xHfalse':id (x + h) = comp f g xlb <= x <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:RHfalse:g (x + h) = g xHfalse':id (x + h) = comp f g xlb <= x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:RHfalse:g (x + h) = g xHfalse':comp f g (x + h) = comp f g xlb <= x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rh'_neq:h' <> 0Rabs (1 / ((f (g x + h') - f (g x)) / h') - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ub1 / (h / (g (x + h) - g x)) = (g (x + h) - g x) / hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:RHfalse:g (x + h) = g xHfalse':h = 0Falsef, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:RHfalse:g (x + h) = g xHfalse':id (x + h) = comp f g xlb <= x <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:RHfalse:g (x + h) = g xHfalse':id (x + h) = comp f g xlb <= x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:RHfalse:g (x + h) = g xHfalse':comp f g (x + h) = comp f g xlb <= x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rh'_neq:h' <> 0Rabs (1 / ((f (g x + h') - f (g x)) / h') - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ub1 / (h / (g (x + h) - g x)) = (g (x + h) - g x) / hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:RHfalse:g (x + h) = g xHfalse':id (x + h) = comp f g xlb <= x <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:RHfalse:g (x + h) = g xHfalse':id (x + h) = comp f g xlb <= x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:RHfalse:g (x + h) = g xHfalse':comp f g (x + h) = comp f g xlb <= x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rh'_neq:h' <> 0Rabs (1 / ((f (g x + h') - f (g x)) / h') - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ub1 / (h / (g (x + h) - g x)) = (g (x + h) - g x) / hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:RHfalse:g (x + h) = g xHfalse':id (x + h) = comp f g xlb <= x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:RHfalse:g (x + h) = g xHfalse':comp f g (x + h) = comp f g xlb <= x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rh'_neq:h' <> 0Rabs (1 / ((f (g x + h') - f (g x)) / h') - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ub1 / (h / (g (x + h) - g x)) = (g (x + h) - g x) / hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:RHfalse:g (x + h) = g xHfalse':comp f g (x + h) = comp f g xlb <= x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rh'_neq:h' <> 0Rabs (1 / ((f (g x + h') - f (g x)) / h') - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ub1 / (h / (g (x + h) - g x)) = (g (x + h) - g x) / hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rh'_neq:h' <> 0Rabs (1 / ((f (g x + h') - f (g x)) / h') - 1 / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ub1 / (h / (g (x + h) - g x)) = (g (x + h) - g x) / hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rh'_neq:h' <> 0Rabs (/ ((f (g x + h') - f (g x)) / h') - / l) < epsf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ub1 / (h / (g (x + h) - g x)) = (g (x + h) - g x) / hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rh'_neq:h' <> 0h' <> 0 /\ Rabs (h' - 0) < alphaf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ub1 / (h / (g (x + h) - g x)) = (g (x + h) - g x) / hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rh'_neq:h' <> 0h' <> 0f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rh'_neq:h' <> 0Rabs (h' - 0) < alphaf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ub1 / (h / (g (x + h) - g x)) = (g (x + h) - g x) / hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rh'_neq:h' <> 0Rabs (h' - 0) < alphaf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ub1 / (h / (g (x + h) - g x)) = (g (x + h) - g x) / hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rh'_neq:h' <> 0Rabs h' < alphaf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ub1 / (h / (g (x + h) - g x)) = (g (x + h) - g x) / hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < alp -> dist R_met (g x0) (g x) < eps0)lb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rh'_neq:h' <> 0Rabs h' < alphaf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ub1 / (h / (g (x + h) - g x)) = (g (x + h) - g x) / hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < alp -> dist R_met (g x0) (g x) < eps0)lb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rh'_neq:h' <> 0forall x0 : R, x0 > 0 /\ (forall x1 : Base R_met, D_x no_cond x x1 /\ dist R_met x1 x < x0 -> dist R_met (g x1) (g x) < mydelta) -> Rabs h' < alphaf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ub1 / (h / (g (x + h) - g x)) = (g (x + h) - g x) / hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < alp -> dist R_met (g x0) (g x) < eps0)lb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rh'_neq:h' <> 0delta3:Rcond3:delta3 > 0 /\ (forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta3 -> dist R_met (g x0) (g x) < mydelta)Rabs h' < alphaf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ub1 / (h / (g (x + h) - g x)) = (g (x + h) - g x) / hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < alp -> dist R_met (g x0) (g x) < eps0)lb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rh'_neq:h' <> 0delta3:Rcond3:delta3 > 0 /\ (forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta3 -> Rabs (g x0 - g x) < mydelta)Rabs h' < alphaf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ub1 / (h / (g (x + h) - g x)) = (g (x + h) - g x) / hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < alp -> dist R_met (g x0) (g x) < eps0)lb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rh'_neq:h' <> 0delta3:Rcond3:delta3 > 0 /\ (forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta3 -> Rabs (g x0 - g x) < mydelta)Rabs (g (x + h) - g x) < alphaf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ub1 / (h / (g (x + h) - g x)) = (g (x + h) - g x) / hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < alp -> dist R_met (g x0) (g x) < eps0)lb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rh'_neq:h' <> 0delta3:Rcond3:delta3 > 0 /\ (forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta3 -> Rabs (g x0 - g x) < mydelta)mydelta <= alphaf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < alp -> dist R_met (g x0) (g x) < eps0)lb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rh'_neq:h' <> 0delta3:Rcond3:delta3 > 0 /\ (forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta3 -> Rabs (g x0 - g x) < mydelta)mydelta_le_alpha:mydelta <= alphaRabs (g (x + h) - g x) < alphaf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ub1 / (h / (g (x + h) - g x)) = (g (x + h) - g x) / hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < alp -> dist R_met (g x0) (g x) < eps0)lb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rh'_neq:h' <> 0delta3:Rcond3:delta3 > 0 /\ (forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta3 -> Rabs (g x0 - g x) < mydelta)delta <= alpha -> delta <= alphaf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < alp -> dist R_met (g x0) (g x) < eps0)lb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rh'_neq:h' <> 0delta3:Rcond3:delta3 > 0 /\ (forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta3 -> Rabs (g x0 - g x) < mydelta)~ delta <= alpha -> alpha <= alphaf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < alp -> dist R_met (g x0) (g x) < eps0)lb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rh'_neq:h' <> 0delta3:Rcond3:delta3 > 0 /\ (forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta3 -> Rabs (g x0 - g x) < mydelta)mydelta_le_alpha:mydelta <= alphaRabs (g (x + h) - g x) < alphaf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ub1 / (h / (g (x + h) - g x)) = (g (x + h) - g x) / hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < alp -> dist R_met (g x0) (g x) < eps0)lb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rh'_neq:h' <> 0delta3:Rcond3:delta3 > 0 /\ (forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta3 -> Rabs (g x0 - g x) < mydelta)~ delta <= alpha -> alpha <= alphaf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < alp -> dist R_met (g x0) (g x) < eps0)lb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rh'_neq:h' <> 0delta3:Rcond3:delta3 > 0 /\ (forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta3 -> Rabs (g x0 - g x) < mydelta)mydelta_le_alpha:mydelta <= alphaRabs (g (x + h) - g x) < alphaf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ub1 / (h / (g (x + h) - g x)) = (g (x + h) - g x) / hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < alp -> dist R_met (g x0) (g x) < eps0)lb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rh'_neq:h' <> 0delta3:Rcond3:delta3 > 0 /\ (forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta3 -> Rabs (g x0 - g x) < mydelta)mydelta_le_alpha:mydelta <= alphaRabs (g (x + h) - g x) < alphaf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ub1 / (h / (g (x + h) - g x)) = (g (x + h) - g x) / hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < alp -> dist R_met (g x0) (g x) < eps0)lb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rh'_neq:h' <> 0delta3:Rcond3:delta3 > 0 /\ (forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta3 -> Rabs (g x0 - g x) < mydelta)mydelta_le_alpha:mydelta <= alphaRabs (g (x + h) - g x) < mydeltaf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < alp -> dist R_met (g x0) (g x) < eps0)lb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rh'_neq:h' <> 0delta3:Rcond3:delta3 > 0 /\ (forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta3 -> Rabs (g x0 - g x) < mydelta)mydelta_le_alpha:mydelta <= alphamydelta <= alphaf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ub1 / (h / (g (x + h) - g x)) = (g (x + h) - g x) / hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < alp -> dist R_met (g x0) (g x) < eps0)lb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta' -> Rabs (g x0 - g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rh'_neq:h' <> 0delta3:Rcond3:delta3 > 0 /\ (forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta3 -> Rabs (g x0 - g x) < mydelta)mydelta_le_alpha:mydelta <= alphaD_x no_cond x (x + h) /\ Rabs (x + h - x) < delta'f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < alp -> dist R_met (g x0) (g x) < eps0)lb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rh'_neq:h' <> 0delta3:Rcond3:delta3 > 0 /\ (forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta3 -> Rabs (g x0 - g x) < mydelta)mydelta_le_alpha:mydelta <= alphamydelta <= alphaf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ub1 / (h / (g (x + h) - g x)) = (g (x + h) - g x) / hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < alp -> dist R_met (g x0) (g x) < eps0)lb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta' -> Rabs (g x0 - g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rh'_neq:h' <> 0delta3:Rcond3:delta3 > 0 /\ (forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta3 -> Rabs (g x0 - g x) < mydelta)mydelta_le_alpha:mydelta <= alphaD_x no_cond x (x + h)f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < alp -> dist R_met (g x0) (g x) < eps0)lb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta' -> Rabs (g x0 - g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rh'_neq:h' <> 0delta3:Rcond3:delta3 > 0 /\ (forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta3 -> Rabs (g x0 - g x) < mydelta)mydelta_le_alpha:mydelta <= alphaRabs (x + h - x) < delta'f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < alp -> dist R_met (g x0) (g x) < eps0)lb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rh'_neq:h' <> 0delta3:Rcond3:delta3 > 0 /\ (forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta3 -> Rabs (g x0 - g x) < mydelta)mydelta_le_alpha:mydelta <= alphamydelta <= alphaf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ub1 / (h / (g (x + h) - g x)) = (g (x + h) - g x) / hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < alp -> dist R_met (g x0) (g x) < eps0)lb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta' -> Rabs (g x0 - g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rh'_neq:h' <> 0delta3:Rcond3:delta3 > 0 /\ (forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta3 -> Rabs (g x0 - g x) < mydelta)mydelta_le_alpha:mydelta <= alphano_cond (x + h) /\ x <> x + hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < alp -> dist R_met (g x0) (g x) < eps0)lb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta' -> Rabs (g x0 - g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rh'_neq:h' <> 0delta3:Rcond3:delta3 > 0 /\ (forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta3 -> Rabs (g x0 - g x) < mydelta)mydelta_le_alpha:mydelta <= alphaRabs (x + h - x) < delta'f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < alp -> dist R_met (g x0) (g x) < eps0)lb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rh'_neq:h' <> 0delta3:Rcond3:delta3 > 0 /\ (forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta3 -> Rabs (g x0 - g x) < mydelta)mydelta_le_alpha:mydelta <= alphamydelta <= alphaf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ub1 / (h / (g (x + h) - g x)) = (g (x + h) - g x) / hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < alp -> dist R_met (g x0) (g x) < eps0)lb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta' -> Rabs (g x0 - g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rh'_neq:h' <> 0delta3:Rcond3:delta3 > 0 /\ (forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta3 -> Rabs (g x0 - g x) < mydelta)mydelta_le_alpha:mydelta <= alphano_cond (x + h)f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < alp -> dist R_met (g x0) (g x) < eps0)lb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta' -> Rabs (g x0 - g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rh'_neq:h' <> 0delta3:Rcond3:delta3 > 0 /\ (forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta3 -> Rabs (g x0 - g x) < mydelta)mydelta_le_alpha:mydelta <= alphax <> x + hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < alp -> dist R_met (g x0) (g x) < eps0)lb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta' -> Rabs (g x0 - g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rh'_neq:h' <> 0delta3:Rcond3:delta3 > 0 /\ (forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta3 -> Rabs (g x0 - g x) < mydelta)mydelta_le_alpha:mydelta <= alphaRabs (x + h - x) < delta'f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < alp -> dist R_met (g x0) (g x) < eps0)lb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rh'_neq:h' <> 0delta3:Rcond3:delta3 > 0 /\ (forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta3 -> Rabs (g x0 - g x) < mydelta)mydelta_le_alpha:mydelta <= alphamydelta <= alphaf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ub1 / (h / (g (x + h) - g x)) = (g (x + h) - g x) / hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < alp -> dist R_met (g x0) (g x) < eps0)lb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta' -> Rabs (g x0 - g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rh'_neq:h' <> 0delta3:Rcond3:delta3 > 0 /\ (forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta3 -> Rabs (g x0 - g x) < mydelta)mydelta_le_alpha:mydelta <= alphax <> x + hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < alp -> dist R_met (g x0) (g x) < eps0)lb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta' -> Rabs (g x0 - g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rh'_neq:h' <> 0delta3:Rcond3:delta3 > 0 /\ (forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta3 -> Rabs (g x0 - g x) < mydelta)mydelta_le_alpha:mydelta <= alphaRabs (x + h - x) < delta'f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < alp -> dist R_met (g x0) (g x) < eps0)lb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rh'_neq:h' <> 0delta3:Rcond3:delta3 > 0 /\ (forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta3 -> Rabs (g x0 - g x) < mydelta)mydelta_le_alpha:mydelta <= alphamydelta <= alphaf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ub1 / (h / (g (x + h) - g x)) = (g (x + h) - g x) / hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < alp -> dist R_met (g x0) (g x) < eps0)lb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta' -> Rabs (g x0 - g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rh'_neq:h' <> 0delta3:Rcond3:delta3 > 0 /\ (forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta3 -> Rabs (g x0 - g x) < mydelta)mydelta_le_alpha:mydelta <= alphaHfalse:x = x + hh = 0f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < alp -> dist R_met (g x0) (g x) < eps0)lb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta' -> Rabs (g x0 - g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rh'_neq:h' <> 0delta3:Rcond3:delta3 > 0 /\ (forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta3 -> Rabs (g x0 - g x) < mydelta)mydelta_le_alpha:mydelta <= alphaRabs (x + h - x) < delta'f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < alp -> dist R_met (g x0) (g x) < eps0)lb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rh'_neq:h' <> 0delta3:Rcond3:delta3 > 0 /\ (forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta3 -> Rabs (g x0 - g x) < mydelta)mydelta_le_alpha:mydelta <= alphamydelta <= alphaf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ub1 / (h / (g (x + h) - g x)) = (g (x + h) - g x) / hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < alp -> dist R_met (g x0) (g x) < eps0)lb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta' -> Rabs (g x0 - g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rh'_neq:h' <> 0delta3:Rcond3:delta3 > 0 /\ (forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta3 -> Rabs (g x0 - g x) < mydelta)mydelta_le_alpha:mydelta <= alphaHfalse:x = x + hx + h = xf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < alp -> dist R_met (g x0) (g x) < eps0)lb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta' -> Rabs (g x0 - g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rh'_neq:h' <> 0delta3:Rcond3:delta3 > 0 /\ (forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta3 -> Rabs (g x0 - g x) < mydelta)mydelta_le_alpha:mydelta <= alphaRabs (x + h - x) < delta'f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < alp -> dist R_met (g x0) (g x) < eps0)lb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rh'_neq:h' <> 0delta3:Rcond3:delta3 > 0 /\ (forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta3 -> Rabs (g x0 - g x) < mydelta)mydelta_le_alpha:mydelta <= alphamydelta <= alphaf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ub1 / (h / (g (x + h) - g x)) = (g (x + h) - g x) / hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < alp -> dist R_met (g x0) (g x) < eps0)lb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta' -> Rabs (g x0 - g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rh'_neq:h' <> 0delta3:Rcond3:delta3 > 0 /\ (forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta3 -> Rabs (g x0 - g x) < mydelta)mydelta_le_alpha:mydelta <= alphaRabs (x + h - x) < delta'f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < alp -> dist R_met (g x0) (g x) < eps0)lb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rh'_neq:h' <> 0delta3:Rcond3:delta3 > 0 /\ (forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta3 -> Rabs (g x0 - g x) < mydelta)mydelta_le_alpha:mydelta <= alphamydelta <= alphaf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ub1 / (h / (g (x + h) - g x)) = (g (x + h) - g x) / hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < alp -> dist R_met (g x0) (g x) < eps0)lb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta' -> Rabs (g x0 - g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rh'_neq:h' <> 0delta3:Rcond3:delta3 > 0 /\ (forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta3 -> Rabs (g x0 - g x) < mydelta)mydelta_le_alpha:mydelta <= alphaRabs h < delta'f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < alp -> dist R_met (g x0) (g x) < eps0)lb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rh'_neq:h' <> 0delta3:Rcond3:delta3 > 0 /\ (forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta3 -> Rabs (g x0 - g x) < mydelta)mydelta_le_alpha:mydelta <= alphamydelta <= alphaf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ub1 / (h / (g (x + h) - g x)) = (g (x + h) - g x) / hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < alp -> dist R_met (g x0) (g x) < eps0)lb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta' -> Rabs (g x0 - g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rh'_neq:h' <> 0delta3:Rcond3:delta3 > 0 /\ (forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta3 -> Rabs (g x0 - g x) < mydelta)mydelta_le_alpha:mydelta <= alphaRabs h < delta''f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < alp -> dist R_met (g x0) (g x) < eps0)lb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta' -> Rabs (g x0 - g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rh'_neq:h' <> 0delta3:Rcond3:delta3 > 0 /\ (forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta3 -> Rabs (g x0 - g x) < mydelta)mydelta_le_alpha:mydelta <= alphadelta'' <= delta'f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < alp -> dist R_met (g x0) (g x) < eps0)lb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rh'_neq:h' <> 0delta3:Rcond3:delta3 > 0 /\ (forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta3 -> Rabs (g x0 - g x) < mydelta)mydelta_le_alpha:mydelta <= alphamydelta <= alphaf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ub1 / (h / (g (x + h) - g x)) = (g (x + h) - g x) / hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < alp -> dist R_met (g x0) (g x) < eps0)lb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta' -> Rabs (g x0 - g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rh'_neq:h' <> 0delta3:Rcond3:delta3 > 0 /\ (forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta3 -> Rabs (g x0 - g x) < mydelta)mydelta_le_alpha:mydelta <= alphadelta'' <= delta'f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < alp -> dist R_met (g x0) (g x) < eps0)lb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rh'_neq:h' <> 0delta3:Rcond3:delta3 > 0 /\ (forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta3 -> Rabs (g x0 - g x) < mydelta)mydelta_le_alpha:mydelta <= alphamydelta <= alphaf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ub1 / (h / (g (x + h) - g x)) = (g (x + h) - g x) / hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, (h0 = 0 -> False) -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < alp -> dist R_met (g x0) (g x) < eps0)lb_lt_ub:lb < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) = 0 -> Falsel:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> (l0 = 0 -> False) -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, (h0 = 0 -> False) -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, (h0 = 0 -> False) -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsPremisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (x0 = 0 -> False) /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l = 0 -> Falsealpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, (x0 = 0 -> False) /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, (h0 = 0 -> False) -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta' -> Rabs (g x0 - g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |}:posrealh:Rh_neq:h = 0 -> Falseh_le_delta':Rabs h < delta''Hrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rh'_neq:h' = 0 -> Falsedelta3:Rmydelta_le_alpha:mydelta <= alphaH0:lb < xH1:x < ubH2:lb <= xH3:x <= ubH4:lb <= x + hH5:x + h <= ubH:delta3 > 0H6:forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta3 -> Rabs (g x0 - g x) < mydeltaH8:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (x0 = 0 -> False) /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)delta'' <= delta'f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < alp -> dist R_met (g x0) (g x) < eps0)lb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rh'_neq:h' <> 0delta3:Rcond3:delta3 > 0 /\ (forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta3 -> Rabs (g x0 - g x) < mydelta)mydelta_le_alpha:mydelta <= alphamydelta <= alphaf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ub1 / (h / (g (x + h) - g x)) = (g (x + h) - g x) / hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, (h0 = 0 -> False) -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < alp -> dist R_met (g x0) (g x) < eps0)lb_lt_ub:lb < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) = 0 -> Falsel:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> (l0 = 0 -> False) -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, (h0 = 0 -> False) -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, (h0 = 0 -> False) -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsPremisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (x0 = 0 -> False) /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l = 0 -> Falsealpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, (x0 = 0 -> False) /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, (h0 = 0 -> False) -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta' -> Rabs (g x0 - g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |}:posrealh:Rh_neq:h = 0 -> Falseh_le_delta':Rabs h < delta''Hrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rh'_neq:h' = 0 -> Falsedelta3:Rmydelta_le_alpha:mydelta <= alphaH0:lb < xH1:x < ubH2:lb <= xH3:x <= ubH4:lb <= x + hH5:x + h <= ubH:delta3 > 0H6:forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta3 -> Rabs (g x0 - g x) < mydeltaH8:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (x0 = 0 -> False) /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)delta'' <= mydelta''f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, (h0 = 0 -> False) -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < alp -> dist R_met (g x0) (g x) < eps0)lb_lt_ub:lb < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) = 0 -> Falsel:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> (l0 = 0 -> False) -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, (h0 = 0 -> False) -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, (h0 = 0 -> False) -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsPremisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (x0 = 0 -> False) /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l = 0 -> Falsealpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, (x0 = 0 -> False) /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, (h0 = 0 -> False) -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta' -> Rabs (g x0 - g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |}:posrealh:Rh_neq:h = 0 -> Falseh_le_delta':Rabs h < delta''Hrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rh'_neq:h' = 0 -> Falsedelta3:Rmydelta_le_alpha:mydelta <= alphaH0:lb < xH1:x < ubH2:lb <= xH3:x <= ubH4:lb <= x + hH5:x + h <= ubH:delta3 > 0H6:forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta3 -> Rabs (g x0 - g x) < mydeltaH8:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (x0 = 0 -> False) /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)mydelta'' <= delta'f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < alp -> dist R_met (g x0) (g x) < eps0)lb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rh'_neq:h' <> 0delta3:Rcond3:delta3 > 0 /\ (forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta3 -> Rabs (g x0 - g x) < mydelta)mydelta_le_alpha:mydelta <= alphamydelta <= alphaf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ub1 / (h / (g (x + h) - g x)) = (g (x + h) - g x) / hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, (h0 = 0 -> False) -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < alp -> dist R_met (g x0) (g x) < eps0)lb_lt_ub:lb < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) = 0 -> Falsel:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> (l0 = 0 -> False) -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, (h0 = 0 -> False) -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, (h0 = 0 -> False) -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsPremisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (x0 = 0 -> False) /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l = 0 -> Falsealpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, (x0 = 0 -> False) /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, (h0 = 0 -> False) -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta' -> Rabs (g x0 - g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |}:posrealh:Rh_neq:h = 0 -> Falseh_le_delta':Rabs h < delta''Hrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rh'_neq:h' = 0 -> Falsedelta3:Rmydelta_le_alpha:mydelta <= alphaH0:lb < xH1:x < ubH2:lb <= xH3:x <= ubH4:lb <= x + hH5:x + h <= ubH:delta3 > 0H6:forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta3 -> Rabs (g x0 - g x) < mydeltaH8:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (x0 = 0 -> False) /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)delta'' = mydelta''f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, (h0 = 0 -> False) -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < alp -> dist R_met (g x0) (g x) < eps0)lb_lt_ub:lb < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) = 0 -> Falsel:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> (l0 = 0 -> False) -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, (h0 = 0 -> False) -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, (h0 = 0 -> False) -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsPremisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (x0 = 0 -> False) /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l = 0 -> Falsealpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, (x0 = 0 -> False) /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, (h0 = 0 -> False) -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta' -> Rabs (g x0 - g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |}:posrealh:Rh_neq:h = 0 -> Falseh_le_delta':Rabs h < delta''Hrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rh'_neq:h' = 0 -> Falsedelta3:Rmydelta_le_alpha:mydelta <= alphaH0:lb < xH1:x < ubH2:lb <= xH3:x <= ubH4:lb <= x + hH5:x + h <= ubH:delta3 > 0H6:forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta3 -> Rabs (g x0 - g x) < mydeltaH8:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (x0 = 0 -> False) /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)mydelta'' <= delta'f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < alp -> dist R_met (g x0) (g x) < eps0)lb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rh'_neq:h' <> 0delta3:Rcond3:delta3 > 0 /\ (forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta3 -> Rabs (g x0 - g x) < mydelta)mydelta_le_alpha:mydelta <= alphamydelta <= alphaf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ub1 / (h / (g (x + h) - g x)) = (g (x + h) - g x) / hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, (h0 = 0 -> False) -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < alp -> dist R_met (g x0) (g x) < eps0)lb_lt_ub:lb < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) = 0 -> Falsel:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> (l0 = 0 -> False) -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, (h0 = 0 -> False) -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, (h0 = 0 -> False) -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsPremisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (x0 = 0 -> False) /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l = 0 -> Falsealpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, (x0 = 0 -> False) /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, (h0 = 0 -> False) -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta' -> Rabs (g x0 - g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |}:posrealh:Rh_neq:h = 0 -> Falseh_le_delta':Rabs h < delta''Hrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rh'_neq:h' = 0 -> Falsedelta3:Rmydelta_le_alpha:mydelta <= alphaH0:lb < xH1:x < ubH2:lb <= xH3:x <= ubH4:lb <= x + hH5:x + h <= ubH:delta3 > 0H6:forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta3 -> Rabs (g x0 - g x) < mydeltaH8:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (x0 = 0 -> False) /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0){| pos := mydelta''; cond_pos := mydelta''_pos |} = mydelta''f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, (h0 = 0 -> False) -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < alp -> dist R_met (g x0) (g x) < eps0)lb_lt_ub:lb < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) = 0 -> Falsel:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> (l0 = 0 -> False) -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, (h0 = 0 -> False) -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, (h0 = 0 -> False) -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsPremisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (x0 = 0 -> False) /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l = 0 -> Falsealpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, (x0 = 0 -> False) /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, (h0 = 0 -> False) -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta' -> Rabs (g x0 - g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |}:posrealh:Rh_neq:h = 0 -> Falseh_le_delta':Rabs h < delta''Hrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rh'_neq:h' = 0 -> Falsedelta3:Rmydelta_le_alpha:mydelta <= alphaH0:lb < xH1:x < ubH2:lb <= xH3:x <= ubH4:lb <= x + hH5:x + h <= ubH:delta3 > 0H6:forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta3 -> Rabs (g x0 - g x) < mydeltaH8:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (x0 = 0 -> False) /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)mydelta'' <= delta'f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < alp -> dist R_met (g x0) (g x) < eps0)lb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rh'_neq:h' <> 0delta3:Rcond3:delta3 > 0 /\ (forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta3 -> Rabs (g x0 - g x) < mydelta)mydelta_le_alpha:mydelta <= alphamydelta <= alphaf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ub1 / (h / (g (x + h) - g x)) = (g (x + h) - g x) / hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, (h0 = 0 -> False) -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < alp -> dist R_met (g x0) (g x) < eps0)lb_lt_ub:lb < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) = 0 -> Falsel:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> (l0 = 0 -> False) -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, (h0 = 0 -> False) -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, (h0 = 0 -> False) -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsPremisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (x0 = 0 -> False) /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l = 0 -> Falsealpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, (x0 = 0 -> False) /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, (h0 = 0 -> False) -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta' -> Rabs (g x0 - g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |}:posrealh:Rh_neq:h = 0 -> Falseh_le_delta':Rabs h < delta''Hrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rh'_neq:h' = 0 -> Falsedelta3:Rmydelta_le_alpha:mydelta <= alphaH0:lb < xH1:x < ubH2:lb <= xH3:x <= ubH4:lb <= x + hH5:x + h <= ubH:delta3 > 0H6:forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta3 -> Rabs (g x0 - g x) < mydeltaH8:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (x0 = 0 -> False) /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)mydelta'' <= delta'f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < alp -> dist R_met (g x0) (g x) < eps0)lb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rh'_neq:h' <> 0delta3:Rcond3:delta3 > 0 /\ (forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta3 -> Rabs (g x0 - g x) < mydelta)mydelta_le_alpha:mydelta <= alphamydelta <= alphaf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ub1 / (h / (g (x + h) - g x)) = (g (x + h) - g x) / hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < alp -> dist R_met (g x0) (g x) < eps0)lb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHrewr:h = comp f g (x + h) - comp f g xh':=g (x + h) - g x:Rh'_neq:h' <> 0delta3:Rcond3:delta3 > 0 /\ (forall x0 : R, D_x no_cond x x0 /\ Rabs (x0 - x) < delta3 -> Rabs (g x0 - g x) < mydelta)mydelta_le_alpha:mydelta <= alphamydelta <= alphaf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ub1 / (h / (g (x + h) - g x)) = (g (x + h) - g x) / hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ub1 / (h / (g (x + h) - g x)) = (g (x + h) - g x) / hf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubh <> 0f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubg (x + h) - g x <> 0f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubg (x + h) - g x <> 0f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHfalse:g (x + h) - g x = 0h = 0f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHfalse:g (x + h) - g x = 0x + h = xf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHfalse:g (x + h) - g x = 0comp f g (x + h) = comp f g xf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHfalse:g (x + h) - g x = 0Hfin:comp f g (x + h) = comp f g xx + h = xf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHfalse:g (x + h) = g xcomp f g (x + h) = comp f g xf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHfalse:g (x + h) - g x = 0Hfin:comp f g (x + h) = comp f g xx + h = xf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHfalse:g (x + h) = g xf (g (x + h)) = f (g x)f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHfalse:g (x + h) - g x = 0Hfin:comp f g (x + h) = comp f g xx + h = xf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHfalse:g (x + h) - g x = 0Hfin:comp f g (x + h) = comp f g xx + h = xf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHfalse:g (x + h) - g x = 0Hfin:id (x + h) = comp f g xx + h = xf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHfalse:g (x + h) - g x = 0Hfin:comp f g (x + h) = comp f g xlb <= x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHfalse:g (x + h) - g x = 0Hfin:id (x + h) = id xx + h = xf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHfalse:g (x + h) - g x = 0Hfin:id (x + h) = comp f g xlb <= x <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHfalse:g (x + h) - g x = 0Hfin:comp f g (x + h) = comp f g xlb <= x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHfalse:g (x + h) - g x = 0Hfin:x + h = xx + h = xf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHfalse:g (x + h) - g x = 0Hfin:id (x + h) = comp f g xlb <= x <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHfalse:g (x + h) - g x = 0Hfin:comp f g (x + h) = comp f g xlb <= x + h <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHfalse:g (x + h) - g x = 0Hfin:id (x + h) = comp f g xlb <= x <= ubf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHfalse:g (x + h) - g x = 0Hfin:comp f g (x + h) = comp f g xlb <= x + h <= ubassumption. Qed.f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> {l0 : R | forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (a + h0) - f a) / h0 - l0) < eps0}g_cont_pur:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0x_encad2:lb <= x <= ubl:RHl:derivable_pt_abs f (g x) leps:Reps_pos:0 < epsy:=g x:RHlinv:forall (f0 : R -> R) (D : R -> Prop) (l0 x0 : R), limit1_in f0 D l0 x0 -> l0 <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x1 : Base R_met, D x1 /\ (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) x1 x0 < alp -> (let (Base, dist, _, _, _, _) as m return (Base m -> Base m -> R) := R_met in dist) (/ f0 x1) (/ l0) < eps0)Hf_deriv:forall eps0 : R, 0 < eps0 -> exists delta0 : posreal, forall h0 : R, h0 <> 0 -> Rabs h0 < delta0 -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < eps0deltatemp:posrealHtemp:forall h0 : R, h0 <> 0 -> Rabs h0 < deltatemp -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsHlinv':(forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)) -> l <> 0 -> forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alp -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < eps0)Premisse:forall eps0 : R, eps0 > 0 -> exists alp : R, alp > 0 /\ (forall x0 : R, (fun h0 : R => h0 <> 0) x0 /\ Rabs (x0 - 0) < alp -> Rabs ((f (y + x0) - f y) / x0 - l) < eps0)Premisse2:l <> 0alpha:Ralpha_pos:alpha > 0inv_cont:forall x0 : R, x0 <> 0 /\ Rabs (x0 - 0) < alpha -> Rabs (/ ((f (y + x0) - f y) / x0) - / l) < epsdelta:posrealf_deriv:forall h0 : R, h0 <> 0 -> Rabs h0 < delta -> Rabs ((f (g x + h0) - f (g x)) / h0 - l) < epsmydelta:=Rmin delta alpha:Rmydelta_pos:mydelta > 0delta':Rdelta'_pos:delta' > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta' -> dist R_met (g x0) (g x) < mydeltamydelta'':=Rmin delta' (Rmin (x - lb) (ub - x)):Rmydelta''_pos:mydelta'' > 0delta'':={| pos := mydelta''; cond_pos := mydelta''_pos |} : posreal:posrealh:Rh_neq:h <> 0h_le_delta':Rabs h < delta''H:lb <= x + h <= ubHfalse:g (x + h) - g x = 0Hfin:comp f g (x + h) = comp f g xlb <= x + h <= ubforall (f g : R -> R) (lb ub x : R) (Prf : forall a : R, g lb <= a <= g ub -> derivable_pt f a), continuity_pt g x -> lb < ub -> lb < x < ub -> forall Prg_incr : g lb <= g x <= g ub, (forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0) -> derive_pt f (g x) (Prf (g x) Prg_incr) <> 0 -> derivable_pt g xforall (f g : R -> R) (lb ub x : R) (Prf : forall a : R, g lb <= a <= g ub -> derivable_pt f a), continuity_pt g x -> lb < ub -> lb < x < ub -> forall Prg_incr : g lb <= g x <= g ub, (forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0) -> derive_pt f (g x) (Prf (g x) Prg_incr) <> 0 -> derivable_pt g xf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> derivable_pt f ag_cont_pt:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0Df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0derivable_pt g xf, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> derivable_pt f ag_cont_pt:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0Df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0{l : R | derivable_pt_lim g x l}apply derivable_pt_lim_recip_interv ; assumption. Qed.f, g:R -> Rlb, ub, x:RPrf:forall a : R, g lb <= a <= g ub -> derivable_pt f ag_cont_pt:continuity_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ubPrg_incr:g lb <= g x <= g ubf_eq_g:forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0Df_neq:derive_pt f (g x) (Prf (g x) Prg_incr) <> 0derivable_pt_lim g x (1 / derive_pt f (g x) (Prf (g x) Prg_incr))forall (f g : R -> R) (lb ub x : R), lb < ub -> f lb < x < f ub -> (forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0) -> (forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub) -> (forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f y) -> (forall a : R, lb <= a <= ub -> derivable_pt f a) -> derivable_pt f (g x)forall (f g : R -> R) (lb ub x : R), lb < ub -> f lb < x < f ub -> (forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0) -> (forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub) -> (forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f y) -> (forall a : R, lb <= a <= ub -> derivable_pt f a) -> derivable_pt f (g x)f, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_ok:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a : R, lb <= a <= ub -> derivable_pt f aderivable_pt f (g x)f, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_ok:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a : R, lb <= a <= ub -> derivable_pt f alb <= g x <= ubf, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_ok:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a : R, lb <= a <= ub -> derivable_pt f aLeft_inv:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0lb <= g x <= ubf, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_ok:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a : R, lb <= a <= ub -> derivable_pt f aLeft_inv:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0comp g f lb <= g x <= ubf, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_ok:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a : R, lb <= a <= ub -> derivable_pt f aLeft_inv:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0comp g f lb = lbf, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_ok:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a : R, lb <= a <= ub -> derivable_pt f aLeft_inv:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0comp g f lb <= g x <= comp g f ubf, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_ok:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a : R, lb <= a <= ub -> derivable_pt f aLeft_inv:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0comp g f ub = ubf, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_ok:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a : R, lb <= a <= ub -> derivable_pt f aLeft_inv:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0comp g f lb = lbf, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_ok:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a : R, lb <= a <= ub -> derivable_pt f aLeft_inv:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0g (f lb) <= g x <= g (f ub)f, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_ok:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a : R, lb <= a <= ub -> derivable_pt f aLeft_inv:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0comp g f ub = ubf, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_ok:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a : R, lb <= a <= ub -> derivable_pt f aLeft_inv:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0comp g f lb = lbf, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_ok:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a : R, lb <= a <= ub -> derivable_pt f aLeft_inv:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0Temp:forall x0 y : R, f lb <= x0 -> x0 < y -> y <= f ub -> g x0 < g yg (f lb) <= g x <= g (f ub)f, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_ok:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a : R, lb <= a <= ub -> derivable_pt f aLeft_inv:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0comp g f ub = ubf, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_ok:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a : R, lb <= a <= ub -> derivable_pt f aLeft_inv:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0comp g f lb = lbf, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_ok:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a : R, lb <= a <= ub -> derivable_pt f aLeft_inv:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0comp g f ub = ubf, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_ok:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a : R, lb <= a <= ub -> derivable_pt f aLeft_inv:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0comp g f lb = lbapply Left_inv ; intuition. Qed.f, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_ok:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a : R, lb <= a <= ub -> derivable_pt f aLeft_inv:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0comp g f lb = lbforall (f g : R -> R) (lb ub x : R) (lb_lt_ub : lb < ub) (x_encad : f lb < x < f ub) (f_eq_g : forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0) (g_wf : forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub) (f_incr : forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f y) (f_derivable : forall a : R, lb <= a <= ub -> derivable_pt f a), derive_pt f (g x) (derivable_pt_recip_interv_prelim1 f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr f_derivable) <> 0 -> derivable_pt g xforall (f g : R -> R) (lb ub x : R) (lb_lt_ub : lb < ub) (x_encad : f lb < x < f ub) (f_eq_g : forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0) (g_wf : forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub) (f_incr : forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f y) (f_derivable : forall a : R, lb <= a <= ub -> derivable_pt f a), derive_pt f (g x) (derivable_pt_recip_interv_prelim1 f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr f_derivable) <> 0 -> derivable_pt g xf, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a : R, lb <= a <= ub -> derivable_pt f aDf_neq:derive_pt f (g x) (derivable_pt_recip_interv_prelim1 f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr f_derivable) <> 0derivable_pt g xf, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a : R, lb <= a <= ub -> derivable_pt f aDf_neq:derive_pt f (g x) (derivable_pt_recip_interv_prelim1 f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr f_derivable) <> 0g (f lb) < g x < g (f ub)f, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a : R, lb <= a <= ub -> derivable_pt f aDf_neq:derive_pt f (g x) (derivable_pt_recip_interv_prelim1 f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr f_derivable) <> 0g_incr:g (f lb) < g x < g (f ub)derivable_pt g xf, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a : R, lb <= a <= ub -> derivable_pt f aDf_neq:derive_pt f (g x) (derivable_pt_recip_interv_prelim1 f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr f_derivable) <> 0Temp:forall x0 y : R, f lb <= x0 -> x0 < y -> y <= f ub -> g x0 < g yg (f lb) < g x < g (f ub)f, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a : R, lb <= a <= ub -> derivable_pt f aDf_neq:derive_pt f (g x) (derivable_pt_recip_interv_prelim1 f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr f_derivable) <> 0g_incr:g (f lb) < g x < g (f ub)derivable_pt g xf, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a : R, lb <= a <= ub -> derivable_pt f aDf_neq:derive_pt f (g x) (derivable_pt_recip_interv_prelim1 f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr f_derivable) = 0 -> FalseTemp:forall x0 y : R, f lb <= x0 -> x0 < y -> y <= f ub -> g x0 < g yf lb < xf, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a : R, lb <= a <= ub -> derivable_pt f aDf_neq:derive_pt f (g x) (derivable_pt_recip_interv_prelim1 f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr f_derivable) = 0 -> FalseTemp:forall x0 y : R, f lb <= x0 -> x0 < y -> y <= f ub -> g x0 < g yx <= f ubf, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a : R, lb <= a <= ub -> derivable_pt f aDf_neq:derive_pt f (g x) (derivable_pt_recip_interv_prelim1 f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr f_derivable) = 0 -> FalseTemp:forall x0 y : R, f lb <= x0 -> x0 < y -> y <= f ub -> g x0 < g yf lb <= xf, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a : R, lb <= a <= ub -> derivable_pt f aDf_neq:derive_pt f (g x) (derivable_pt_recip_interv_prelim1 f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr f_derivable) = 0 -> FalseTemp:forall x0 y : R, f lb <= x0 -> x0 < y -> y <= f ub -> g x0 < g yx < f ubf, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a : R, lb <= a <= ub -> derivable_pt f aDf_neq:derive_pt f (g x) (derivable_pt_recip_interv_prelim1 f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr f_derivable) <> 0g_incr:g (f lb) < g x < g (f ub)derivable_pt g xf, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a : R, lb <= a <= ub -> derivable_pt f aDf_neq:derive_pt f (g x) (derivable_pt_recip_interv_prelim1 f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr f_derivable) = 0 -> FalseTemp:forall x0 y : R, f lb <= x0 -> x0 < y -> y <= f ub -> g x0 < g yx <= f ubf, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a : R, lb <= a <= ub -> derivable_pt f aDf_neq:derive_pt f (g x) (derivable_pt_recip_interv_prelim1 f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr f_derivable) = 0 -> FalseTemp:forall x0 y : R, f lb <= x0 -> x0 < y -> y <= f ub -> g x0 < g yf lb <= xf, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a : R, lb <= a <= ub -> derivable_pt f aDf_neq:derive_pt f (g x) (derivable_pt_recip_interv_prelim1 f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr f_derivable) = 0 -> FalseTemp:forall x0 y : R, f lb <= x0 -> x0 < y -> y <= f ub -> g x0 < g yx < f ubf, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a : R, lb <= a <= ub -> derivable_pt f aDf_neq:derive_pt f (g x) (derivable_pt_recip_interv_prelim1 f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr f_derivable) <> 0g_incr:g (f lb) < g x < g (f ub)derivable_pt g xf, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a : R, lb <= a <= ub -> derivable_pt f aDf_neq:derive_pt f (g x) (derivable_pt_recip_interv_prelim1 f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr f_derivable) = 0 -> FalseTemp:forall x0 y : R, f lb <= x0 -> x0 < y -> y <= f ub -> g x0 < g yf lb <= xf, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a : R, lb <= a <= ub -> derivable_pt f aDf_neq:derive_pt f (g x) (derivable_pt_recip_interv_prelim1 f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr f_derivable) = 0 -> FalseTemp:forall x0 y : R, f lb <= x0 -> x0 < y -> y <= f ub -> g x0 < g yx < f ubf, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a : R, lb <= a <= ub -> derivable_pt f aDf_neq:derive_pt f (g x) (derivable_pt_recip_interv_prelim1 f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr f_derivable) <> 0g_incr:g (f lb) < g x < g (f ub)derivable_pt g xf, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a : R, lb <= a <= ub -> derivable_pt f aDf_neq:derive_pt f (g x) (derivable_pt_recip_interv_prelim1 f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr f_derivable) = 0 -> FalseTemp:forall x0 y : R, f lb <= x0 -> x0 < y -> y <= f ub -> g x0 < g yx < f ubf, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a : R, lb <= a <= ub -> derivable_pt f aDf_neq:derive_pt f (g x) (derivable_pt_recip_interv_prelim1 f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr f_derivable) <> 0g_incr:g (f lb) < g x < g (f ub)derivable_pt g xf, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a : R, lb <= a <= ub -> derivable_pt f aDf_neq:derive_pt f (g x) (derivable_pt_recip_interv_prelim1 f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr f_derivable) <> 0g_incr:g (f lb) < g x < g (f ub)derivable_pt g xf, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a : R, lb <= a <= ub -> derivable_pt f aDf_neq:derive_pt f (g x) (derivable_pt_recip_interv_prelim1 f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr f_derivable) <> 0g_incr:g (f lb) < g x < g (f ub)g (f lb) <= g x <= g (f ub)f, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a : R, lb <= a <= ub -> derivable_pt f aDf_neq:derive_pt f (g x) (derivable_pt_recip_interv_prelim1 f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr f_derivable) <> 0g_incr:g (f lb) < g x < g (f ub)g_incr2:g (f lb) <= g x <= g (f ub)derivable_pt g xf, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a : R, lb <= a <= ub -> derivable_pt f aDf_neq:derive_pt f (g x) (derivable_pt_recip_interv_prelim1 f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr f_derivable) <> 0g_incr:g (f lb) < g x < g (f ub)g_incr2:g (f lb) <= g x <= g (f ub)derivable_pt g xf, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a : R, lb <= a <= ub -> derivable_pt f aDf_neq:derive_pt f (g x) (derivable_pt_recip_interv_prelim1 f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr f_derivable) <> 0g_incr:g (f lb) < g x < g (f ub)g_incr2:g (f lb) <= g x <= g (f ub)g_eq_f:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0derivable_pt g xf, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a : R, lb <= a <= ub -> derivable_pt f aDf_neq:derive_pt f (g x) (derivable_pt_recip_interv_prelim1 f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr f_derivable) <> 0g_incr:g (f lb) < g x < g (f ub)g_incr2:g (f lb) <= g x <= g (f ub)g_eq_f:forall x0 : R, lb <= x0 <= ub -> g (f x0) = x0derivable_pt g xf, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a : R, lb <= a <= ub -> derivable_pt f aDf_neq:derive_pt f (g x) (derivable_pt_recip_interv_prelim1 f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr f_derivable) <> 0g_incr:g (f lb) < g x < g (f ub)g_incr2:g (f lb) <= g x <= g (f ub)g_eq_f:forall x0 : R, lb <= x0 <= ub -> g (f x0) = x0forall a : R, g (f lb) <= a <= g (f ub) -> derivable_pt f af, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a : R, lb <= a <= ub -> derivable_pt f aDf_neq:derive_pt f (g x) (derivable_pt_recip_interv_prelim1 f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr f_derivable) <> 0g_incr:g (f lb) < g x < g (f ub)g_incr2:g (f lb) <= g x <= g (f ub)g_eq_f:forall x0 : R, lb <= x0 <= ub -> g (f x0) = x0f_derivable2:forall a : R, g (f lb) <= a <= g (f ub) -> derivable_pt f aderivable_pt g xf, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a0 : R, lb <= a0 <= ub -> derivable_pt f a0Df_neq:derive_pt f (g x) (derivable_pt_recip_interv_prelim1 f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr f_derivable) <> 0g_incr:g (f lb) < g x < g (f ub)g_incr2:g (f lb) <= g x <= g (f ub)g_eq_f:forall x0 : R, lb <= x0 <= ub -> g (f x0) = x0a:Ra_encad:g (f lb) <= a <= g (f ub)lb <= a <= ubf, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a : R, lb <= a <= ub -> derivable_pt f aDf_neq:derive_pt f (g x) (derivable_pt_recip_interv_prelim1 f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr f_derivable) <> 0g_incr:g (f lb) < g x < g (f ub)g_incr2:g (f lb) <= g x <= g (f ub)g_eq_f:forall x0 : R, lb <= x0 <= ub -> g (f x0) = x0f_derivable2:forall a : R, g (f lb) <= a <= g (f ub) -> derivable_pt f aderivable_pt g xf, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a : R, lb <= a <= ub -> derivable_pt f aDf_neq:derive_pt f (g x) (derivable_pt_recip_interv_prelim1 f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr f_derivable) <> 0g_incr:g (f lb) < g x < g (f ub)g_incr2:g (f lb) <= g x <= g (f ub)g_eq_f:forall x0 : R, lb <= x0 <= ub -> g (f x0) = x0f_derivable2:forall a : R, g (f lb) <= a <= g (f ub) -> derivable_pt f aderivable_pt g xf, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a : R, lb <= a <= ub -> derivable_pt f aDf_neq:derive_pt f (g x) (derivable_pt_recip_interv_prelim1 f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr f_derivable) <> 0g_incr:g (f lb) < g x < g (f ub)g_incr2:g (f lb) <= g x <= g (f ub)g_eq_f:forall x0 : R, lb <= x0 <= ub -> g (f x0) = x0f_derivable2:forall a : R, g (f lb) <= a <= g (f ub) -> derivable_pt f acontinuity_pt g xf, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a : R, lb <= a <= ub -> derivable_pt f aDf_neq:derive_pt f (g x) (derivable_pt_recip_interv_prelim1 f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr f_derivable) <> 0g_incr:g (f lb) < g x < g (f ub)g_incr2:g (f lb) <= g x <= g (f ub)g_eq_f:forall x0 : R, lb <= x0 <= ub -> g (f x0) = x0f_derivable2:forall a : R, g (f lb) <= a <= g (f ub) -> derivable_pt f af lb < f ubf, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a : R, lb <= a <= ub -> derivable_pt f aDf_neq:derive_pt f (g x) (derivable_pt_recip_interv_prelim1 f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr f_derivable) <> 0g_incr:g (f lb) < g x < g (f ub)g_incr2:g (f lb) <= g x <= g (f ub)g_eq_f:forall x0 : R, lb <= x0 <= ub -> g (f x0) = x0f_derivable2:forall a : R, g (f lb) <= a <= g (f ub) -> derivable_pt f af lb < x < f ubf, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a : R, lb <= a <= ub -> derivable_pt f aDf_neq:derive_pt f (g x) (derivable_pt_recip_interv_prelim1 f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr f_derivable) <> 0g_incr:g (f lb) < g x < g (f ub)g_incr2:g (f lb) <= g x <= g (f ub)g_eq_f:forall x0 : R, lb <= x0 <= ub -> g (f x0) = x0f_derivable2:forall a : R, g (f lb) <= a <= g (f ub) -> derivable_pt f aforall x0 : R, f lb <= x0 <= f ub -> comp f g x0 = id x0f, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a : R, lb <= a <= ub -> derivable_pt f aDf_neq:derive_pt f (g x) (derivable_pt_recip_interv_prelim1 f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr f_derivable) <> 0g_incr:g (f lb) < g x < g (f ub)g_incr2:g (f lb) <= g x <= g (f ub)g_eq_f:forall x0 : R, lb <= x0 <= ub -> g (f x0) = x0f_derivable2:forall a : R, g (f lb) <= a <= g (f ub) -> derivable_pt f aderive_pt f (g x) (f_derivable2 (g x) g_incr2) <> 0f, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a0 : R, lb <= a0 <= ub -> derivable_pt f a0Df_neq:derive_pt f (g x) (derivable_pt_recip_interv_prelim1 f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr f_derivable) = 0 -> Falseg_eq_f:forall x0 : R, lb <= x0 <= ub -> g (f x0) = x0f_derivable2:forall a0 : R, g (f lb) <= a0 <= g (f ub) -> derivable_pt f a0H:g (f lb) < g xH0:g x < g (f ub)H1:g (f lb) <= g xH2:g x <= g (f ub)a:RH4:lb <= aH5:a <= ubcontinuity_pt f af, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a : R, lb <= a <= ub -> derivable_pt f aDf_neq:derive_pt f (g x) (derivable_pt_recip_interv_prelim1 f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr f_derivable) = 0 -> Falseg_eq_f:forall x0 : R, lb <= x0 <= ub -> g (f x0) = x0f_derivable2:forall a : R, g (f lb) <= a <= g (f ub) -> derivable_pt f aH:g (f lb) < g xH0:g x < g (f ub)H1:g (f lb) <= g xH2:g x <= g (f ub)f lb < xf, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a : R, lb <= a <= ub -> derivable_pt f aDf_neq:derive_pt f (g x) (derivable_pt_recip_interv_prelim1 f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr f_derivable) = 0 -> Falseg_eq_f:forall x0 : R, lb <= x0 <= ub -> g (f x0) = x0f_derivable2:forall a : R, g (f lb) <= a <= g (f ub) -> derivable_pt f aH:g (f lb) < g xH0:g x < g (f ub)H1:g (f lb) <= g xH2:g x <= g (f ub)x < f ubf, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a : R, lb <= a <= ub -> derivable_pt f aDf_neq:derive_pt f (g x) (derivable_pt_recip_interv_prelim1 f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr f_derivable) <> 0g_incr:g (f lb) < g x < g (f ub)g_incr2:g (f lb) <= g x <= g (f ub)g_eq_f:forall x0 : R, lb <= x0 <= ub -> g (f x0) = x0f_derivable2:forall a : R, g (f lb) <= a <= g (f ub) -> derivable_pt f af lb < f ubf, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a : R, lb <= a <= ub -> derivable_pt f aDf_neq:derive_pt f (g x) (derivable_pt_recip_interv_prelim1 f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr f_derivable) <> 0g_incr:g (f lb) < g x < g (f ub)g_incr2:g (f lb) <= g x <= g (f ub)g_eq_f:forall x0 : R, lb <= x0 <= ub -> g (f x0) = x0f_derivable2:forall a : R, g (f lb) <= a <= g (f ub) -> derivable_pt f af lb < x < f ubf, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a : R, lb <= a <= ub -> derivable_pt f aDf_neq:derive_pt f (g x) (derivable_pt_recip_interv_prelim1 f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr f_derivable) <> 0g_incr:g (f lb) < g x < g (f ub)g_incr2:g (f lb) <= g x <= g (f ub)g_eq_f:forall x0 : R, lb <= x0 <= ub -> g (f x0) = x0f_derivable2:forall a : R, g (f lb) <= a <= g (f ub) -> derivable_pt f aforall x0 : R, f lb <= x0 <= f ub -> comp f g x0 = id x0f, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a : R, lb <= a <= ub -> derivable_pt f aDf_neq:derive_pt f (g x) (derivable_pt_recip_interv_prelim1 f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr f_derivable) <> 0g_incr:g (f lb) < g x < g (f ub)g_incr2:g (f lb) <= g x <= g (f ub)g_eq_f:forall x0 : R, lb <= x0 <= ub -> g (f x0) = x0f_derivable2:forall a : R, g (f lb) <= a <= g (f ub) -> derivable_pt f aderive_pt f (g x) (f_derivable2 (g x) g_incr2) <> 0f, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a : R, lb <= a <= ub -> derivable_pt f aDf_neq:derive_pt f (g x) (derivable_pt_recip_interv_prelim1 f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr f_derivable) = 0 -> Falseg_eq_f:forall x0 : R, lb <= x0 <= ub -> g (f x0) = x0f_derivable2:forall a : R, g (f lb) <= a <= g (f ub) -> derivable_pt f aH:g (f lb) < g xH0:g x < g (f ub)H1:g (f lb) <= g xH2:g x <= g (f ub)f lb < xf, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a : R, lb <= a <= ub -> derivable_pt f aDf_neq:derive_pt f (g x) (derivable_pt_recip_interv_prelim1 f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr f_derivable) = 0 -> Falseg_eq_f:forall x0 : R, lb <= x0 <= ub -> g (f x0) = x0f_derivable2:forall a : R, g (f lb) <= a <= g (f ub) -> derivable_pt f aH:g (f lb) < g xH0:g x < g (f ub)H1:g (f lb) <= g xH2:g x <= g (f ub)x < f ubf, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a : R, lb <= a <= ub -> derivable_pt f aDf_neq:derive_pt f (g x) (derivable_pt_recip_interv_prelim1 f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr f_derivable) <> 0g_incr:g (f lb) < g x < g (f ub)g_incr2:g (f lb) <= g x <= g (f ub)g_eq_f:forall x0 : R, lb <= x0 <= ub -> g (f x0) = x0f_derivable2:forall a : R, g (f lb) <= a <= g (f ub) -> derivable_pt f af lb < f ubf, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a : R, lb <= a <= ub -> derivable_pt f aDf_neq:derive_pt f (g x) (derivable_pt_recip_interv_prelim1 f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr f_derivable) <> 0g_incr:g (f lb) < g x < g (f ub)g_incr2:g (f lb) <= g x <= g (f ub)g_eq_f:forall x0 : R, lb <= x0 <= ub -> g (f x0) = x0f_derivable2:forall a : R, g (f lb) <= a <= g (f ub) -> derivable_pt f af lb < x < f ubf, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a : R, lb <= a <= ub -> derivable_pt f aDf_neq:derive_pt f (g x) (derivable_pt_recip_interv_prelim1 f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr f_derivable) <> 0g_incr:g (f lb) < g x < g (f ub)g_incr2:g (f lb) <= g x <= g (f ub)g_eq_f:forall x0 : R, lb <= x0 <= ub -> g (f x0) = x0f_derivable2:forall a : R, g (f lb) <= a <= g (f ub) -> derivable_pt f aforall x0 : R, f lb <= x0 <= f ub -> comp f g x0 = id x0f, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a : R, lb <= a <= ub -> derivable_pt f aDf_neq:derive_pt f (g x) (derivable_pt_recip_interv_prelim1 f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr f_derivable) <> 0g_incr:g (f lb) < g x < g (f ub)g_incr2:g (f lb) <= g x <= g (f ub)g_eq_f:forall x0 : R, lb <= x0 <= ub -> g (f x0) = x0f_derivable2:forall a : R, g (f lb) <= a <= g (f ub) -> derivable_pt f aderive_pt f (g x) (f_derivable2 (g x) g_incr2) <> 0f, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a : R, lb <= a <= ub -> derivable_pt f aDf_neq:derive_pt f (g x) (derivable_pt_recip_interv_prelim1 f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr f_derivable) = 0 -> Falseg_eq_f:forall x0 : R, lb <= x0 <= ub -> g (f x0) = x0f_derivable2:forall a : R, g (f lb) <= a <= g (f ub) -> derivable_pt f aH:g (f lb) < g xH0:g x < g (f ub)H1:g (f lb) <= g xH2:g x <= g (f ub)x < f ubf, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a : R, lb <= a <= ub -> derivable_pt f aDf_neq:derive_pt f (g x) (derivable_pt_recip_interv_prelim1 f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr f_derivable) <> 0g_incr:g (f lb) < g x < g (f ub)g_incr2:g (f lb) <= g x <= g (f ub)g_eq_f:forall x0 : R, lb <= x0 <= ub -> g (f x0) = x0f_derivable2:forall a : R, g (f lb) <= a <= g (f ub) -> derivable_pt f af lb < f ubf, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a : R, lb <= a <= ub -> derivable_pt f aDf_neq:derive_pt f (g x) (derivable_pt_recip_interv_prelim1 f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr f_derivable) <> 0g_incr:g (f lb) < g x < g (f ub)g_incr2:g (f lb) <= g x <= g (f ub)g_eq_f:forall x0 : R, lb <= x0 <= ub -> g (f x0) = x0f_derivable2:forall a : R, g (f lb) <= a <= g (f ub) -> derivable_pt f af lb < x < f ubf, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a : R, lb <= a <= ub -> derivable_pt f aDf_neq:derive_pt f (g x) (derivable_pt_recip_interv_prelim1 f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr f_derivable) <> 0g_incr:g (f lb) < g x < g (f ub)g_incr2:g (f lb) <= g x <= g (f ub)g_eq_f:forall x0 : R, lb <= x0 <= ub -> g (f x0) = x0f_derivable2:forall a : R, g (f lb) <= a <= g (f ub) -> derivable_pt f aforall x0 : R, f lb <= x0 <= f ub -> comp f g x0 = id x0f, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a : R, lb <= a <= ub -> derivable_pt f aDf_neq:derive_pt f (g x) (derivable_pt_recip_interv_prelim1 f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr f_derivable) <> 0g_incr:g (f lb) < g x < g (f ub)g_incr2:g (f lb) <= g x <= g (f ub)g_eq_f:forall x0 : R, lb <= x0 <= ub -> g (f x0) = x0f_derivable2:forall a : R, g (f lb) <= a <= g (f ub) -> derivable_pt f aderive_pt f (g x) (f_derivable2 (g x) g_incr2) <> 0f, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a : R, lb <= a <= ub -> derivable_pt f aDf_neq:derive_pt f (g x) (derivable_pt_recip_interv_prelim1 f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr f_derivable) <> 0g_incr:g (f lb) < g x < g (f ub)g_incr2:g (f lb) <= g x <= g (f ub)g_eq_f:forall x0 : R, lb <= x0 <= ub -> g (f x0) = x0f_derivable2:forall a : R, g (f lb) <= a <= g (f ub) -> derivable_pt f af lb < f ubf, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a : R, lb <= a <= ub -> derivable_pt f aDf_neq:derive_pt f (g x) (derivable_pt_recip_interv_prelim1 f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr f_derivable) <> 0g_incr:g (f lb) < g x < g (f ub)g_incr2:g (f lb) <= g x <= g (f ub)g_eq_f:forall x0 : R, lb <= x0 <= ub -> g (f x0) = x0f_derivable2:forall a : R, g (f lb) <= a <= g (f ub) -> derivable_pt f af lb < x < f ubf, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a : R, lb <= a <= ub -> derivable_pt f aDf_neq:derive_pt f (g x) (derivable_pt_recip_interv_prelim1 f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr f_derivable) <> 0g_incr:g (f lb) < g x < g (f ub)g_incr2:g (f lb) <= g x <= g (f ub)g_eq_f:forall x0 : R, lb <= x0 <= ub -> g (f x0) = x0f_derivable2:forall a : R, g (f lb) <= a <= g (f ub) -> derivable_pt f aforall x0 : R, f lb <= x0 <= f ub -> comp f g x0 = id x0f, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a : R, lb <= a <= ub -> derivable_pt f aDf_neq:derive_pt f (g x) (derivable_pt_recip_interv_prelim1 f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr f_derivable) <> 0g_incr:g (f lb) < g x < g (f ub)g_incr2:g (f lb) <= g x <= g (f ub)g_eq_f:forall x0 : R, lb <= x0 <= ub -> g (f x0) = x0f_derivable2:forall a : R, g (f lb) <= a <= g (f ub) -> derivable_pt f aderive_pt f (g x) (f_derivable2 (g x) g_incr2) <> 0f, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a : R, lb <= a <= ub -> derivable_pt f aDf_neq:derive_pt f (g x) (derivable_pt_recip_interv_prelim1 f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr f_derivable) <> 0g_incr:g (f lb) < g x < g (f ub)g_incr2:g (f lb) <= g x <= g (f ub)g_eq_f:forall x0 : R, lb <= x0 <= ub -> g (f x0) = x0f_derivable2:forall a : R, g (f lb) <= a <= g (f ub) -> derivable_pt f af lb < x < f ubf, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a : R, lb <= a <= ub -> derivable_pt f aDf_neq:derive_pt f (g x) (derivable_pt_recip_interv_prelim1 f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr f_derivable) <> 0g_incr:g (f lb) < g x < g (f ub)g_incr2:g (f lb) <= g x <= g (f ub)g_eq_f:forall x0 : R, lb <= x0 <= ub -> g (f x0) = x0f_derivable2:forall a : R, g (f lb) <= a <= g (f ub) -> derivable_pt f aforall x0 : R, f lb <= x0 <= f ub -> comp f g x0 = id x0f, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a : R, lb <= a <= ub -> derivable_pt f aDf_neq:derive_pt f (g x) (derivable_pt_recip_interv_prelim1 f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr f_derivable) <> 0g_incr:g (f lb) < g x < g (f ub)g_incr2:g (f lb) <= g x <= g (f ub)g_eq_f:forall x0 : R, lb <= x0 <= ub -> g (f x0) = x0f_derivable2:forall a : R, g (f lb) <= a <= g (f ub) -> derivable_pt f aderive_pt f (g x) (f_derivable2 (g x) g_incr2) <> 0f, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a : R, lb <= a <= ub -> derivable_pt f aDf_neq:derive_pt f (g x) (derivable_pt_recip_interv_prelim1 f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr f_derivable) <> 0g_incr:g (f lb) < g x < g (f ub)g_incr2:g (f lb) <= g x <= g (f ub)g_eq_f:forall x0 : R, lb <= x0 <= ub -> g (f x0) = x0f_derivable2:forall a : R, g (f lb) <= a <= g (f ub) -> derivable_pt f aforall x0 : R, f lb <= x0 <= f ub -> comp f g x0 = id x0f, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a : R, lb <= a <= ub -> derivable_pt f aDf_neq:derive_pt f (g x) (derivable_pt_recip_interv_prelim1 f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr f_derivable) <> 0g_incr:g (f lb) < g x < g (f ub)g_incr2:g (f lb) <= g x <= g (f ub)g_eq_f:forall x0 : R, lb <= x0 <= ub -> g (f x0) = x0f_derivable2:forall a : R, g (f lb) <= a <= g (f ub) -> derivable_pt f aderive_pt f (g x) (f_derivable2 (g x) g_incr2) <> 0rewrite pr_nu_var2_interv with (g:=f) (lb:=lb) (ub:=ub) (pr2:=derivable_pt_recip_interv_prelim1 f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr f_derivable) ; [| |rewrite g_eq_f in g_incr ; rewrite g_eq_f in g_incr| ] ; intuition. Qed. (****************************************************)f, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0g_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yf_derivable:forall a : R, lb <= a <= ub -> derivable_pt f aDf_neq:derive_pt f (g x) (derivable_pt_recip_interv_prelim1 f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr f_derivable) <> 0g_incr:g (f lb) < g x < g (f ub)g_incr2:g (f lb) <= g x <= g (f ub)g_eq_f:forall x0 : R, lb <= x0 <= ub -> g (f x0) = x0f_derivable2:forall a : R, g (f lb) <= a <= g (f ub) -> derivable_pt f aderive_pt f (g x) (f_derivable2 (g x) g_incr2) <> 0
(****************************************************)forall (f g : R -> R) (lb ub x : R) (Prf : derivable_pt f (g x)) (Prg : derivable_pt g x), lb < ub -> lb < x < ub -> (forall x0 : R, lb < x0 < ub -> comp f g x0 = id x0) -> derive_pt f (g x) Prf <> 0 -> derive_pt g x Prg = 1 / derive_pt f (g x) Prfforall (f g : R -> R) (lb ub x : R) (Prf : derivable_pt f (g x)) (Prg : derivable_pt g x), lb < ub -> lb < x < ub -> (forall x0 : R, lb < x0 < ub -> comp f g x0 = id x0) -> derive_pt f (g x) Prf <> 0 -> derive_pt g x Prg = 1 / derive_pt f (g x) Prff, g:R -> Rlb, ub, x:RPrf:derivable_pt f (g x)Prg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_recip:forall x0 : R, lb < x0 < ub -> comp f g x0 = id x0Df_neq:derive_pt f (g x) Prf <> 0derive_pt g x Prg = 1 / derive_pt f (g x) Prff, g:R -> Rlb, ub, x:RPrf:derivable_pt f (g x)Prg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_recip:forall x0 : R, lb < x0 < ub -> comp f g x0 = id x0Df_neq:derive_pt f (g x) Prf <> 0derive_pt g x Prg * derive_pt f (g x) Prf * / derive_pt f (g x) Prf = 1 / derive_pt f (g x) Prff, g:R -> Rlb, ub, x:RPrf:derivable_pt f (g x)Prg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_recip:forall x0 : R, lb < x0 < ub -> comp f g x0 = id x0Df_neq:derive_pt f (g x) Prf <> 0derive_pt g x Prg * derive_pt f (g x) Prf * / derive_pt f (g x) Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f (g x)Prg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_recip:forall x0 : R, lb < x0 < ub -> comp f g x0 = id x0Df_neq:derive_pt f (g x) Prf <> 0derive_pt g x Prg * derive_pt f (g x) Prf * / derive_pt f (g x) Prf = 1 * / derive_pt f (g x) Prff, g:R -> Rlb, ub, x:RPrf:derivable_pt f (g x)Prg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_recip:forall x0 : R, lb < x0 < ub -> comp f g x0 = id x0Df_neq:derive_pt f (g x) Prf <> 0derive_pt g x Prg * derive_pt f (g x) Prf * / derive_pt f (g x) Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f (g x)Prg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_recip:forall x0 : R, lb < x0 < ub -> comp f g x0 = id x0Df_neq:derive_pt f (g x) Prf <> 0/ derive_pt f (g x) Prf * (derive_pt g x Prg * derive_pt f (g x) Prf) = 1 * / derive_pt f (g x) Prff, g:R -> Rlb, ub, x:RPrf:derivable_pt f (g x)Prg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_recip:forall x0 : R, lb < x0 < ub -> comp f g x0 = id x0Df_neq:derive_pt f (g x) Prf <> 0derive_pt g x Prg * derive_pt f (g x) Prf * / derive_pt f (g x) Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f (g x)Prg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_recip:forall x0 : R, lb < x0 < ub -> comp f g x0 = id x0Df_neq:derive_pt f (g x) Prf <> 0/ derive_pt f (g x) Prf * (derive_pt g x Prg * derive_pt f (g x) Prf) = / derive_pt f (g x) Prf * 1f, g:R -> Rlb, ub, x:RPrf:derivable_pt f (g x)Prg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_recip:forall x0 : R, lb < x0 < ub -> comp f g x0 = id x0Df_neq:derive_pt f (g x) Prf <> 0derive_pt g x Prg * derive_pt f (g x) Prf * / derive_pt f (g x) Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f (g x)Prg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_recip:forall x0 : R, lb < x0 < ub -> comp f g x0 = id x0Df_neq:derive_pt f (g x) Prf <> 0derive_pt g x Prg * derive_pt f (g x) Prf = 1f, g:R -> Rlb, ub, x:RPrf:derivable_pt f (g x)Prg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_recip:forall x0 : R, lb < x0 < ub -> comp f g x0 = id x0Df_neq:derive_pt f (g x) Prf <> 0derive_pt g x Prg * derive_pt f (g x) Prf * / derive_pt f (g x) Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f (g x)Prg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_recip:forall x0 : R, lb < x0 < ub -> comp f g x0 = id x0Df_neq:derive_pt f (g x) Prf <> 0derive_pt f (g x) Prf * derive_pt g x Prg = 1f, g:R -> Rlb, ub, x:RPrf:derivable_pt f (g x)Prg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_recip:forall x0 : R, lb < x0 < ub -> comp f g x0 = id x0Df_neq:derive_pt f (g x) Prf <> 0derive_pt g x Prg * derive_pt f (g x) Prf * / derive_pt f (g x) Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f (g x)Prg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_recip:forall x0 : R, lb < x0 < ub -> comp f g x0 = id x0Df_neq:derive_pt f (g x) Prf <> 0derive_pt (comp f g) x (derivable_pt_comp g f x Prg Prf) = 1f, g:R -> Rlb, ub, x:RPrf:derivable_pt f (g x)Prg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_recip:forall x0 : R, lb < x0 < ub -> comp f g x0 = id x0Df_neq:derive_pt f (g x) Prf <> 0derive_pt g x Prg * derive_pt f (g x) Prf * / derive_pt f (g x) Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f (g x)Prg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_recip:forall x0 : R, lb < x0 < ub -> comp f g x0 = id x0Df_neq:derive_pt f (g x) Prf <> 0x_encad2:lb <= x <= ubderive_pt (comp f g) x (derivable_pt_comp g f x Prg Prf) = 1f, g:R -> Rlb, ub, x:RPrf:derivable_pt f (g x)Prg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_recip:forall x0 : R, lb < x0 < ub -> comp f g x0 = id x0Df_neq:derive_pt f (g x) Prf <> 0derive_pt g x Prg * derive_pt f (g x) Prf * / derive_pt f (g x) Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f (g x)Prg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_recip:forall x0 : R, lb < x0 < ub -> comp f g x0 = id x0Df_neq:derive_pt f (g x) Prf <> 0derive_pt g x Prg * derive_pt f (g x) Prf * / derive_pt f (g x) Prf = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f (g x)Prg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_recip:forall x0 : R, lb < x0 < ub -> comp f g x0 = id x0Df_neq:derive_pt f (g x) Prf <> 0derive_pt g x Prg * 1 = derive_pt g x Prgf, g:R -> Rlb, ub, x:RPrf:derivable_pt f (g x)Prg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_recip:forall x0 : R, lb < x0 < ub -> comp f g x0 = id x0Df_neq:derive_pt f (g x) Prf <> 0derive_pt f (g x) Prf <> 0assumption. Qed.f, g:R -> Rlb, ub, x:RPrf:derivable_pt f (g x)Prg:derivable_pt g xlb_lt_ub:lb < ubx_encad:lb < x < ublocal_recip:forall x0 : R, lb < x0 < ub -> comp f g x0 = id x0Df_neq:derive_pt f (g x) Prf <> 0derive_pt f (g x) Prf <> 0forall (f g : R -> R) (lb ub x : R), lb < ub -> f lb < x < f ub -> (forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f y) -> (forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub) -> (forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0) -> lb < g x < ubforall (f g : R -> R) (lb ub x : R), lb < ub -> f lb < x < f ub -> (forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f y) -> (forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub) -> (forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0) -> lb < g x < ubf, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0lb < g x < ubf, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0Temp:forall x0 y : R, f lb <= x0 -> x0 < y -> y <= f ub -> g x0 < g ylb < g x < ubf, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0Temp:forall x0 y : R, f lb <= x0 -> x0 < y -> y <= f ub -> g x0 < g yLeft_inv:forall x0 : R, lb <= x0 <= ub -> comp g f x0 = id x0lb < g x < ubf, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0Temp:forall x0 y : R, f lb <= x0 -> x0 < y -> y <= f ub -> g x0 < g yLeft_inv:forall x0 : R, lb <= x0 <= ub -> g (f x0) = x0lb < g x < ubf, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0Temp:forall x0 y : R, f lb <= x0 -> x0 < y -> y <= f ub -> g x0 < g yLeft_inv:forall x0 : R, lb <= x0 <= ub -> g (f x0) = x0g (f lb) < g xf, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0Temp:forall x0 y : R, f lb <= x0 -> x0 < y -> y <= f ub -> g x0 < g yLeft_inv:forall x0 : R, lb <= x0 <= ub -> g (f x0) = x0lb <= lb <= ubf, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0Temp:forall x0 y : R, f lb <= x0 -> x0 < y -> y <= f ub -> g x0 < g yLeft_inv:forall x0 : R, lb <= x0 <= ub -> g (f x0) = x0g x < g (f ub)f, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0Temp:forall x0 y : R, f lb <= x0 -> x0 < y -> y <= f ub -> g x0 < g yLeft_inv:forall x0 : R, lb <= x0 <= ub -> g (f x0) = x0lb <= ub <= ubf, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0Temp:forall x0 y : R, f lb <= x0 -> x0 < y -> y <= f ub -> g x0 < g yLeft_inv:forall x0 : R, lb <= x0 <= ub -> g (f x0) = x0lb <= lb <= ubf, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0Temp:forall x0 y : R, f lb <= x0 -> x0 < y -> y <= f ub -> g x0 < g yLeft_inv:forall x0 : R, lb <= x0 <= ub -> g (f x0) = x0g x < g (f ub)f, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0Temp:forall x0 y : R, f lb <= x0 -> x0 < y -> y <= f ub -> g x0 < g yLeft_inv:forall x0 : R, lb <= x0 <= ub -> g (f x0) = x0lb <= ub <= ubf, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0Temp:forall x0 y : R, f lb <= x0 -> x0 < y -> y <= f ub -> g x0 < g yLeft_inv:forall x0 : R, lb <= x0 <= ub -> g (f x0) = x0g x < g (f ub)f, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0Temp:forall x0 y : R, f lb <= x0 -> x0 < y -> y <= f ub -> g x0 < g yLeft_inv:forall x0 : R, lb <= x0 <= ub -> g (f x0) = x0lb <= ub <= ubintuition. Qed.f, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0Temp:forall x0 y : R, f lb <= x0 -> x0 < y -> y <= f ub -> g x0 < g yLeft_inv:forall x0 : R, lb <= x0 <= ub -> g (f x0) = x0lb <= ub <= ubforall (f g : R -> R) (lb ub x : R), lb < ub -> f lb < x < f ub -> (forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f y) -> (forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub) -> (forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0) -> lb <= g x <= ubforall (f g : R -> R) (lb ub x : R), lb < ub -> f lb < x < f ub -> (forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f y) -> (forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub) -> (forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0) -> lb <= g x <= ubf, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0lb <= g x <= ubsplit ; apply Rlt_le ; intuition. Qed.f, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubf_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0Temp:lb < g x < ublb <= g x <= ubforall (f g : R -> R) (lb ub x : R) (lb_lt_ub : lb < ub) (x_encad : f lb < x < f ub) (f_incr : forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f y) (g_wf : forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub) (Prf : forall a : R, lb <= a <= ub -> derivable_pt f a) (f_eq_g : forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0) (Df_neq : derive_pt f (g x) (derivable_pt_recip_interv_prelim1 f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr Prf) <> 0), derive_pt g x (derivable_pt_recip_interv f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr Prf Df_neq) = 1 / derive_pt f (g x) (Prf (g x) (derive_pt_recip_interv_prelim1_1 f g lb ub x lb_lt_ub x_encad f_incr g_wf f_eq_g))forall (f g : R -> R) (lb ub x : R) (lb_lt_ub : lb < ub) (x_encad : f lb < x < f ub) (f_incr : forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f y) (g_wf : forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub) (Prf : forall a : R, lb <= a <= ub -> derivable_pt f a) (f_eq_g : forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0) (Df_neq : derive_pt f (g x) (derivable_pt_recip_interv_prelim1 f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr Prf) <> 0), derive_pt g x (derivable_pt_recip_interv f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr Prf Df_neq) = 1 / derive_pt f (g x) (Prf (g x) (derive_pt_recip_interv_prelim1_1 f g lb ub x lb_lt_ub x_encad f_incr g_wf f_eq_g))f, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubPrf:forall a : R, lb <= a <= ub -> derivable_pt f af_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0Df_neq:derive_pt f (g x) (derivable_pt_recip_interv_prelim1 f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr Prf) <> 0derive_pt g x (derivable_pt_recip_interv f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr Prf Df_neq) = 1 / derive_pt f (g x) (Prf (g x) (derive_pt_recip_interv_prelim1_1 f g lb ub x lb_lt_ub x_encad f_incr g_wf f_eq_g))f, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubPrf:forall a : R, lb <= a <= ub -> derivable_pt f af_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0Df_neq:derive_pt f (g x) (derivable_pt_recip_interv_prelim1 f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr Prf) <> 0g_incr:lb <= g x <= ubderive_pt g x (derivable_pt_recip_interv f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr Prf Df_neq) = 1 / derive_pt f (g x) (Prf (g x) (derive_pt_recip_interv_prelim1_1 f g lb ub x lb_lt_ub x_encad f_incr g_wf f_eq_g))f, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubPrf:forall a : R, lb <= a <= ub -> derivable_pt f af_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0Df_neq:derive_pt f (g x) (derivable_pt_recip_interv_prelim1 f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr Prf) <> 0g_incr:lb <= g x <= ubderive_pt f (g x) (Prf (g x) (derive_pt_recip_interv_prelim1_1 f g lb ub x lb_lt_ub x_encad f_incr g_wf f_eq_g)) <> 0f, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubPrf:forall a : R, lb <= a <= ub -> derivable_pt f af_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0Df_neq:derive_pt f (g x) (derivable_pt_recip_interv_prelim1 f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr Prf) <> 0g_incr:lb <= g x <= ubHfalse:derive_pt f (g x) (Prf (g x) (derive_pt_recip_interv_prelim1_1 f g lb ub x lb_lt_ub x_encad f_incr g_wf f_eq_g)) = 0derive_pt f (g x) (derivable_pt_recip_interv_prelim1 f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr Prf) = 0exact (derive_pt_recip_interv_prelim1_0 f g lb ub x lb_lt_ub x_encad f_incr g_wf f_eq_g). Qed. (****************************************************)f, g:R -> Rlb, ub, x:Rlb_lt_ub:lb < ubx_encad:f lb < x < f ubf_incr:forall x0 y : R, lb <= x0 -> x0 < y -> y <= ub -> f x0 < f yg_wf:forall x0 : R, f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ubPrf:forall a : R, lb <= a <= ub -> derivable_pt f af_eq_g:forall x0 : R, f lb <= x0 -> x0 <= f ub -> comp f g x0 = id x0Df_neq:derive_pt f (g x) (derivable_pt_recip_interv_prelim1 f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr Prf) <> 0g_incr:lb <= g x <= ubHfalse:derive_pt f (g x) (Prf (g x) (derive_pt_recip_interv_prelim1_1 f g lb ub x lb_lt_ub x_encad f_incr g_wf f_eq_g)) = 0lb < g x < ub
(****************************************************) (* begin hide *)forall x ub lb : R, lb < x -> x < ub -> 0 < (ub - lb) / 2forall x ub lb : R, lb < x -> x < ub -> 0 < (ub - lb) / 2lra. Qed.x, ub, lb:Rlb_lt_x:lb < xx_lt_ub:x < ub0 < (ub - lb) / 2apply (mkposreal ((ub-lb)/2) (ub_lt_2_pos x ub lb lb_lt_x x_lt_ub)). Defined. (* end hide *)x, lb, ub:Rlb_lt_x:lb < xx_lt_ub:x < ubposrealforall (fn fn' : nat -> R -> R) (f g : R -> R) (x c : R) (r : posreal), Boule c r x -> (forall (y : R) (n : nat), Boule c r y -> derivable_pt_lim (fn n) y (fn' n y)) -> (forall y : R, Boule c r y -> Un_cv (fun n : nat => fn n y) (f y)) -> CVU fn' g c r -> (forall y : R, Boule c r y -> continuity_pt g y) -> derivable_pt_lim f x (g x)forall (fn fn' : nat -> R -> R) (f g : R -> R) (x c : R) (r : posreal), Boule c r x -> (forall (y : R) (n : nat), Boule c r y -> derivable_pt_lim (fn n) y (fn' n y)) -> (forall y : R, Boule c r y -> Un_cv (fun n : nat => fn n y) (f y)) -> CVU fn' g c r -> (forall y : R, Boule c r y -> continuity_pt g y) -> derivable_pt_lim f x (g x)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn_CV_f:forall y : R, Boule c' r y -> Un_cv (fun n : nat => fn n y) (f y)fn'_CVU_g:CVU fn' g c' rg_cont:forall y : R, Boule c' r y -> continuity_pt g yeps:Reps_pos:0 < epsexists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn_CV_f:forall y : R, Boule c' r y -> Un_cv (fun n : nat => fn n y) (f y)fn'_CVU_g:CVU fn' g c' rg_cont:forall y : R, Boule c' r y -> continuity_pt g yeps:Reps_pos:0 < epseps_8_pos:0 < eps / 8exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn_CV_f:forall y : R, Boule c' r y -> Un_cv (fun n : nat => fn n y) (f y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8exists delta : posreal, forall h : R, h <> 0 -> Rabs h < delta -> Rabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn_CV_f:forall y : R, Boule c' r y -> Un_cv (fun n : nat => fn n y) (f y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yexists delta0 : posreal, forall h : R, h <> 0 -> Rabs h < delta0 -> Rabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn_CV_f:forall y : R, Boule c' r y -> Un_cv (fun n : nat => fn n y) (f y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn_CV_f:forall y : R, Boule c' r y -> Un_cv (fun n : nat => fn n y) (f y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < delta0 < Rabs h * eps / 4fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn_CV_f:forall y : R, Boule c' r y -> Un_cv (fun n : nat => fn n y) (f y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4Rabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn_CV_f:forall y : R, Boule c' r y -> Un_cv (fun n : nat => fn n y) (f y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < delta0 < Rabs hfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn_CV_f:forall y : R, Boule c' r y -> Un_cv (fun n : nat => fn n y) (f y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < delta0 < eps * / 4fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn_CV_f:forall y : R, Boule c' r y -> Un_cv (fun n : nat => fn n y) (f y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4Rabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn_CV_f:forall y : R, Boule c' r y -> Un_cv (fun n : nat => fn n y) (f y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < delta0 < eps * / 4fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn_CV_f:forall y : R, Boule c' r y -> Un_cv (fun n : nat => fn n y) (f y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4Rabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn_CV_f:forall y : R, Boule c' r y -> Un_cv (fun n : nat => fn n y) (f y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4Rabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn_CV_f:forall y : R, Boule c' r y -> Un_cv (fun n : nat => fn n y) (f y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4Rabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn_CV_f:forall y : R, Boule c' r y -> Un_cv (fun n : nat => fn n y) (f y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4Boule x delta (x + h)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn_CV_f:forall y : R, Boule c' r y -> Un_cv (fun n : nat => fn n y) (f y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)Rabs ((f (x + h) - f x) / h - g x) < epsx:Rdelta:posrealh:Rhinbdelta:h < delta /\ - delta < hRabs (x + h - x) < deltafn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn_CV_f:forall y : R, Boule c' r y -> Un_cv (fun n : nat => fn n y) (f y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)Rabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn_CV_f:forall y : R, Boule c' r y -> Un_cv (fun n : nat => fn n y) (f y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)Rabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn_CV_f:forall y : R, Boule c' r y -> Un_cv (fun n : nat => fn n y) (f y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)Boule c' r (x + h)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn_CV_f:forall y : R, Boule c' r y -> Un_cv (fun n : nat => fn n y) (f y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)t:Boule c' r (x + h)Rabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn_CV_f:forall y : R, Boule c' r y -> Un_cv (fun n : nat => fn n y) (f y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)t:Boule c' r (x + h)Rabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn_CV_f:forall y : R, Boule c' r y -> Un_cv (fun n : nat => fn n y) (f y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)t:Boule c' r (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4Rabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4Rabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8Rabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natRabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natRabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) <= Rabs (f (x + h) - fn N (x + h) - (f x - fn N x)) + Rabs (fn N (x + h) - fn N x - h * g x)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natRabs (f (x + h) - fn N (x + h) - (f x - fn N x)) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natRabs (f (x + h) - fn N (x + h) - (f x - fn N x)) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natRabs (f (x + h) - fn N (x + h) - (f x - fn N x)) + Rabs (fn N (x + h) - fn N x - h * g x) <= Rabs (f (x + h) - fn N (x + h)) + Rabs (- (f x - fn N x)) + Rabs (fn N (x + h) - fn N x - h * g x)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natRabs (f (x + h) - fn N (x + h)) + Rabs (- (f x - fn N x)) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natRabs (f (x + h) - fn N (x + h)) + Rabs (- (f x - fn N x)) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0Rabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0forall c : R, x + h < c < x -> derivable_pt (fn N) cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c : R, x + h < c < x -> derivable_pt (fn N) cRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0c:Rc_encad:x + h < c < x{l : R | derivable_pt_abs (fn N) c l}fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c : R, x + h < c < x -> derivable_pt (fn N) cRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0c:Rc_encad:x + h < c < xBoule c' r cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c : R, x + h < c < x -> derivable_pt (fn N) cRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0c:Rc_encad:x + h < c < xBoule x delta cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0c:Rc_encad:x + h < c < xt:Boule x delta cBoule c' r cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c : R, x + h < c < x -> derivable_pt (fn N) cRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4H:x + h - x < deltaH0:- delta < x + h - xN1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0c:RH1:x + h < cH2:c < xBoule x delta cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0c:Rc_encad:x + h < c < xt:Boule x delta cBoule c' r cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c : R, x + h < c < x -> derivable_pt (fn N) cRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0c:Rc_encad:x + h < c < xt:Boule x delta cBoule c' r cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c : R, x + h < c < x -> derivable_pt (fn N) cRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c : R, x + h < c < x -> derivable_pt (fn N) cRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c : R, x + h < c < x -> derivable_pt (fn N) cforall c : R, x + h < c < x -> derivable_pt id cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c : R, x + h < c < x -> derivable_pt (fn N) cpr2:forall c : R, x + h < c < x -> derivable_pt id cRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c : R, x + h < c < x -> derivable_pt (fn N) cpr2:forall c : R, x + h < c < x -> derivable_pt id cRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c : R, x + h < c < x -> derivable_pt (fn N) cpr2:forall c : R, x + h < c < x -> derivable_pt id cxh_x:x + h < xRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c : R, x + h < c < x -> derivable_pt (fn N) cpr2:forall c : R, x + h < c < x -> derivable_pt id cxh_x:x + h < xforall c : R, x + h <= c <= x -> continuity_pt (fn N) cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c : R, x + h < c < x -> derivable_pt (fn N) cpr2:forall c : R, x + h < c < x -> derivable_pt id cxh_x:x + h < xpr3:forall c : R, x + h <= c <= x -> continuity_pt (fn N) cRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xc:Rc_encad:x + h <= c <= xderivable_pt (fn N) cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c : R, x + h < c < x -> derivable_pt (fn N) cpr2:forall c : R, x + h < c < x -> derivable_pt id cxh_x:x + h < xpr3:forall c : R, x + h <= c <= x -> continuity_pt (fn N) cRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h = 0 -> Falsehinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xc:RH:x + h <= cH0:c <= xBoule c' r cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c : R, x + h < c < x -> derivable_pt (fn N) cpr2:forall c : R, x + h < c < x -> derivable_pt id cxh_x:x + h < xpr3:forall c : R, x + h <= c <= x -> continuity_pt (fn N) cRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h = 0 -> Falsehinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xc:RH:x + h <= cH0:c <= xBoule x delta cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h = 0 -> Falsehinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xc:RH:x + h <= cH0:c <= xt:Boule x delta cBoule c' r cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c : R, x + h < c < x -> derivable_pt (fn N) cpr2:forall c : R, x + h < c < x -> derivable_pt id cxh_x:x + h < xpr3:forall c : R, x + h <= c <= x -> continuity_pt (fn N) cRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h = 0 -> Falsehinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4H1:x + h - x < deltaH2:- delta < x + h - xN1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xc:RH:x + h <= cH0:c <= xBoule x delta cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h = 0 -> Falsehinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xc:RH:x + h <= cH0:c <= xt:Boule x delta cBoule c' r cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c : R, x + h < c < x -> derivable_pt (fn N) cpr2:forall c : R, x + h < c < x -> derivable_pt id cxh_x:x + h < xpr3:forall c : R, x + h <= c <= x -> continuity_pt (fn N) cRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h = 0 -> Falsehinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xc:RH:x + h <= cH0:c <= xt:Boule x delta cBoule c' r cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c : R, x + h < c < x -> derivable_pt (fn N) cpr2:forall c : R, x + h < c < x -> derivable_pt id cxh_x:x + h < xpr3:forall c : R, x + h <= c <= x -> continuity_pt (fn N) cRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c : R, x + h < c < x -> derivable_pt (fn N) cpr2:forall c : R, x + h < c < x -> derivable_pt id cxh_x:x + h < xpr3:forall c : R, x + h <= c <= x -> continuity_pt (fn N) cRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c : R, x + h < c < x -> derivable_pt (fn N) cpr2:forall c : R, x + h < c < x -> derivable_pt id cxh_x:x + h < xpr3:forall c : R, x + h <= c <= x -> continuity_pt (fn N) cforall c : R, x + h <= c <= x -> continuity_pt id cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c : R, x + h < c < x -> derivable_pt (fn N) cpr2:forall c : R, x + h < c < x -> derivable_pt id cxh_x:x + h < xpr3:forall c : R, x + h <= c <= x -> continuity_pt (fn N) cpr4:forall c : R, x + h <= c <= x -> continuity_pt id cRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c : R, x + h < c < x -> derivable_pt (fn N) cpr2:forall c : R, x + h < c < x -> derivable_pt id cxh_x:x + h < xpr3:forall c : R, x + h <= c <= x -> continuity_pt (fn N) cpr4:forall c : R, x + h <= c <= x -> continuity_pt id cRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xHc:(id x - id (x + h)) * derive_pt (fn N) c (pr1 c P) = (fn N x - fn N (x + h)) * derive_pt id c (pr2 c P)Rabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xHc:(id x - id (x + h)) * derive_pt (fn N) c (pr1 c P) = (fn N x - fn N (x + h)) * derive_pt id c (pr2 c P)h * derive_pt (fn N) c (pr1 c P) = fn N (x + h) - fn N xfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xHc:(id x - id (x + h)) * derive_pt (fn N) c (pr1 c P) = (fn N x - fn N (x + h)) * derive_pt id c (pr2 c P)Hc':h * derive_pt (fn N) c (pr1 c P) = fn N (x + h) - fn N xRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xHc:(id x - id (x + h)) * derive_pt (fn N) c (pr1 c P) = (fn N x - fn N (x + h)) * derive_pt id c (pr2 c P)-1 * (h * derive_pt (fn N) c (pr1 c P)) = -1 * (fn N (x + h) - fn N x)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xHc:(id x - id (x + h)) * derive_pt (fn N) c (pr1 c P) = (fn N x - fn N (x + h)) * derive_pt id c (pr2 c P)-1 <> 0fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xHc:(id x - id (x + h)) * derive_pt (fn N) c (pr1 c P) = (fn N x - fn N (x + h)) * derive_pt id c (pr2 c P)Hc':h * derive_pt (fn N) c (pr1 c P) = fn N (x + h) - fn N xRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xHc:(id x - id (x + h)) * derive_pt (fn N) c (pr1 c P) = (fn N x - fn N (x + h)) * derive_pt id c (pr2 c P)- h * derive_pt (fn N) c (pr1 c P) = -1 * (fn N (x + h) - fn N x)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xHc:(id x - id (x + h)) * derive_pt (fn N) c (pr1 c P) = (fn N x - fn N (x + h)) * derive_pt id c (pr2 c P)-1 <> 0fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xHc:(id x - id (x + h)) * derive_pt (fn N) c (pr1 c P) = (fn N x - fn N (x + h)) * derive_pt id c (pr2 c P)Hc':h * derive_pt (fn N) c (pr1 c P) = fn N (x + h) - fn N xRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xHc:(id x - id (x + h)) * derive_pt (fn N) c (pr1 c P) = (fn N x - fn N (x + h)) * derive_pt id c (pr2 c P)- h * derive_pt (fn N) c (pr1 c P) = - (fn N (x + h) - fn N x)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xHc:(id x - id (x + h)) * derive_pt (fn N) c (pr1 c P) = (fn N x - fn N (x + h)) * derive_pt id c (pr2 c P)-1 <> 0fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xHc:(id x - id (x + h)) * derive_pt (fn N) c (pr1 c P) = (fn N x - fn N (x + h)) * derive_pt id c (pr2 c P)Hc':h * derive_pt (fn N) c (pr1 c P) = fn N (x + h) - fn N xRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xHc:(id x - id (x + h)) * derive_pt (fn N) c (pr1 c P) = (fn N x - fn N (x + h)) * derive_pt id c (pr2 c P)(id x - id (x + h)) * derive_pt (fn N) c (pr1 c P) = - (fn N (x + h) - fn N x)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xHc:(id x - id (x + h)) * derive_pt (fn N) c (pr1 c P) = (fn N x - fn N (x + h)) * derive_pt id c (pr2 c P)-1 <> 0fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xHc:(id x - id (x + h)) * derive_pt (fn N) c (pr1 c P) = (fn N x - fn N (x + h)) * derive_pt id c (pr2 c P)Hc':h * derive_pt (fn N) c (pr1 c P) = fn N (x + h) - fn N xRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xHc:(id x - id (x + h)) * derive_pt (fn N) c (pr1 c P) = (fn N x - fn N (x + h)) * derive_pt id c (pr2 c P)(id x - id (x + h)) * derive_pt (fn N) c (pr1 c P) = - (fn N (x + h) - fn N x) * derive_pt id c (pr2 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xHc:(id x - id (x + h)) * derive_pt (fn N) c (pr1 c P) = (fn N x - fn N (x + h)) * derive_pt id c (pr2 c P)-1 <> 0fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xHc:(id x - id (x + h)) * derive_pt (fn N) c (pr1 c P) = (fn N x - fn N (x + h)) * derive_pt id c (pr2 c P)Hc':h * derive_pt (fn N) c (pr1 c P) = fn N (x + h) - fn N xRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xHc:(id x - id (x + h)) * derive_pt (fn N) c (pr1 c P) = (fn N x - fn N (x + h)) * derive_pt id c (pr2 c P)(id x - id (x + h)) * derive_pt (fn N) c (pr1 c P) = (fn N x - fn N (x + h)) * derive_pt id c (pr2 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xHc:(id x - id (x + h)) * derive_pt (fn N) c (pr1 c P) = (fn N x - fn N (x + h)) * derive_pt id c (pr2 c P)-1 <> 0fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xHc:(id x - id (x + h)) * derive_pt (fn N) c (pr1 c P) = (fn N x - fn N (x + h)) * derive_pt id c (pr2 c P)Hc':h * derive_pt (fn N) c (pr1 c P) = fn N (x + h) - fn N xRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xHc:(id x - id (x + h)) * derive_pt (fn N) c (pr1 c P) = (fn N x - fn N (x + h)) * derive_pt id c (pr2 c P)-1 <> 0fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xHc:(id x - id (x + h)) * derive_pt (fn N) c (pr1 c P) = (fn N x - fn N (x + h)) * derive_pt id c (pr2 c P)Hc':h * derive_pt (fn N) c (pr1 c P) = fn N (x + h) - fn N xRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xHc:(id x - id (x + h)) * derive_pt (fn N) c (pr1 c P) = (fn N x - fn N (x + h)) * derive_pt id c (pr2 c P)Hc':h * derive_pt (fn N) c (pr1 c P) = fn N (x + h) - fn N xRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (h * derive_pt (fn N) c (pr1 c P) - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (h * fn' N c - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (h * (fn' N c - g x)) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs h * Rabs (fn' N c - g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs h * Rabs (fn' N c - g x) < Rabs h * eps / 4 + Rabs (f x - fn N x) + Rabs h * Rabs (fn' N c - g x)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xRabs h * eps / 4 + Rabs (f x - fn N x) + Rabs h * Rabs (fn' N c - g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> Rabs (fn n (x + h) - f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < x(N >= N1)%natfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xRabs h * eps / 4 + Rabs (f x - fn N x) + Rabs h * Rabs (fn' N c - g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xRabs h * eps / 4 + Rabs (f x - fn N x) + Rabs h * Rabs (fn' N c - g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xRabs h * eps / 4 + Rabs (f x - fn N x) + Rabs h * Rabs (fn' N c - g x) < Rabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * Rabs (fn' N c - g x)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xRabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * Rabs (fn' N c - g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xRabs (f x - fn N x) < Rabs h * eps / 4fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xRabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * Rabs (fn' N c - g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> Rabs (fn n x - f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < x(N >= N2)%natfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xRabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * Rabs (fn' N c - g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xRabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * Rabs (fn' N c - g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xRabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * Rabs (fn' N c - g c + (g c - g x)) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xRabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * Rabs (fn' N c - g c + (g c - g x)) <= Rabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * Rabs (fn' N c - g c) + Rabs h * Rabs (g c - g x)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xRabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * Rabs (fn' N c - g c) + Rabs h * Rabs (g c - g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < x0 <= Rabs hfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xRabs (fn' N c - g c + (g c - g x)) <= Rabs (fn' N c - g c) + Rabs (g c - g x)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xRabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * Rabs (fn' N c - g c) + Rabs h * Rabs (g c - g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xRabs (fn' N c - g c + (g c - g x)) <= Rabs (fn' N c - g c) + Rabs (g c - g x)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xRabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * Rabs (fn' N c - g c) + Rabs h * Rabs (g c - g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xRabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * Rabs (fn' N c - g c) + Rabs h * Rabs (g c - g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xRabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * Rabs (fn' N c - g c) + Rabs h * Rabs (g c - g x) < Rabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * (eps / 8) + Rabs h * Rabs (g c - g x)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xRabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * (eps / 8) + Rabs h * Rabs (g c - g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < x0 < Rabs hfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xRabs (fn' N c - g c) < eps / 8fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xRabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * (eps / 8) + Rabs h * Rabs (g c - g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xRabs (fn' N c - g c) < eps / 8fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xRabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * (eps / 8) + Rabs h * Rabs (g c - g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < x(N3 <= N)%natfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xBoule c' r cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xRabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * (eps / 8) + Rabs h * Rabs (g c - g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xBoule c' r cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xRabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * (eps / 8) + Rabs h * Rabs (g c - g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xBoule x delta cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xt:Boule x delta cBoule c' r cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xRabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * (eps / 8) + Rabs h * Rabs (g c - g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RH:x + h < cH0:c < xBoule x delta cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xt:Boule x delta cBoule c' r cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xRabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * (eps / 8) + Rabs h * Rabs (g c - g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4H1:x + h - x < deltaH2:- delta < x + h - xN1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RH:x + h < cH0:c < xBoule x delta cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xt:Boule x delta cBoule c' r cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xRabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * (eps / 8) + Rabs h * Rabs (g c - g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xt:Boule x delta cBoule c' r cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xRabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * (eps / 8) + Rabs h * Rabs (g c - g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xRabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * (eps / 8) + Rabs h * Rabs (g c - g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xRabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * (eps / 8) + Rabs h * Rabs (g c - g x) < Rabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * (eps / 8) + Rabs h * (eps / 8)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xRabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * (eps / 8) + Rabs h * (eps / 8) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < x0 < Rabs hfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xeps / 8 + Rabs (g c - g x) < eps / 8 + eps / 8fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xRabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * (eps / 8) + Rabs h * (eps / 8) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xeps / 8 + Rabs (g c - g x) < eps / 8 + eps / 8fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xRabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * (eps / 8) + Rabs h * (eps / 8) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : R, D_x no_cond x x0 /\ R_dist x0 x < delta1 -> R_dist (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xno_cond cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : R, D_x no_cond x x0 /\ R_dist x0 x < delta1 -> R_dist (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xx <> cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : R, D_x no_cond x x0 /\ R_dist x0 x < delta1 -> R_dist (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xR_dist c x < delta1fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xRabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * (eps / 8) + Rabs h * (eps / 8) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : R, D_x no_cond x x0 /\ R_dist x0 x < delta1 -> R_dist (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xx <> cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : R, D_x no_cond x x0 /\ R_dist x0 x < delta1 -> R_dist (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xR_dist c x < delta1fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xRabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * (eps / 8) + Rabs h * (eps / 8) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : R, D_x no_cond x x0 /\ R_dist x0 x < delta1 -> R_dist (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xR_dist c x < delta1fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xRabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * (eps / 8) + Rabs h * (eps / 8) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : R, D_x no_cond x x0 /\ R_dist x0 x < delta1 -> R_dist (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xR_dist c x < Rabs hfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : R, D_x no_cond x x0 /\ R_dist x0 x < delta1 -> R_dist (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xRabs h < delta1fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xRabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * (eps / 8) + Rabs h * (eps / 8) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : R, D_x no_cond x x0 /\ R_dist x0 x < delta1 -> R_dist (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xc - x < Rabs hfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : R, D_x no_cond x x0 /\ R_dist x0 x < delta1 -> R_dist (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < x- Rabs h < c - xfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : R, D_x no_cond x x0 /\ R_dist x0 x < delta1 -> R_dist (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xRabs h < delta1fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xRabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * (eps / 8) + Rabs h * (eps / 8) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : R, D_x no_cond x x0 /\ R_dist x0 x < delta1 -> R_dist (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xc - x < 0fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : R, D_x no_cond x x0 /\ R_dist x0 x < delta1 -> R_dist (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < x0 < Rabs hfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : R, D_x no_cond x x0 /\ R_dist x0 x < delta1 -> R_dist (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < x- Rabs h < c - xfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : R, D_x no_cond x x0 /\ R_dist x0 x < delta1 -> R_dist (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xRabs h < delta1fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xRabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * (eps / 8) + Rabs h * (eps / 8) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : R, D_x no_cond x x0 /\ R_dist x0 x < delta1 -> R_dist (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < x0 < Rabs hfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : R, D_x no_cond x x0 /\ R_dist x0 x < delta1 -> R_dist (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < x- Rabs h < c - xfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : R, D_x no_cond x x0 /\ R_dist x0 x < delta1 -> R_dist (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xRabs h < delta1fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xRabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * (eps / 8) + Rabs h * (eps / 8) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : R, D_x no_cond x x0 /\ R_dist x0 x < delta1 -> R_dist (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < x- Rabs h < c - xfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : R, D_x no_cond x x0 /\ R_dist x0 x < delta1 -> R_dist (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xRabs h < delta1fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xRabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * (eps / 8) + Rabs h * (eps / 8) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : R, D_x no_cond x x0 /\ R_dist x0 x < delta1 -> R_dist (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xh < c - xfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : R, D_x no_cond x x0 /\ R_dist x0 x < delta1 -> R_dist (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xRabs h < delta1fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xRabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * (eps / 8) + Rabs h * (eps / 8) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : R, D_x no_cond x x0 /\ R_dist x0 x < delta1 -> R_dist (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xRabs h < delta1fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xRabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * (eps / 8) + Rabs h * (eps / 8) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsx, c':Rr:posrealdelta1:Rdelta1_pos:delta1 > 0delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rxhinbxdelta:Boule x delta (x + h)Rabs h < delta1fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xRabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * (eps / 8) + Rabs h * (eps / 8) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsx, c':Rr:posrealdelta1:Rdelta1_pos:delta1 > 0delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:RP':Boule x {| pos := delta1; cond_pos := delta1_pos |} (x + h)Rabs h < delta1fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xRabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * (eps / 8) + Rabs h * (eps / 8) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xRabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * (eps / 8) + Rabs h * (eps / 8) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xRabs h * eps / 4 + (Rabs h * eps / 4 + Rabs h * (eps / 8 + eps / 8)) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xRabs h * (eps / 4 + eps / 4 + eps / 8 + eps / 8) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < x0 < Rabs hfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xeps / 4 + eps / 4 + eps / 8 + eps / 8 < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xeps / 4 + eps / 4 + eps / 8 + eps / 8 < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xl:RHl:derivable_pt_abs (fn N) c lfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xl:RHl:derivable_pt_abs (fn N) c ll = fn' N cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xl:RHl:derivable_pt_abs (fn N) c lTemp:l = fn' N cfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xl:RHl:derivable_pt_abs (fn N) c lBoule c' r cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xl:RHl:derivable_pt_abs (fn N) c lbc'rc:Boule c' r cl = fn' N cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xl:RHl:derivable_pt_abs (fn N) c lTemp:l = fn' N cfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xl:RHl:derivable_pt_abs (fn N) c lBoule x delta cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xl:RHl:derivable_pt_abs (fn N) c lt:Boule x delta cBoule c' r cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xl:RHl:derivable_pt_abs (fn N) c lbc'rc:Boule c' r cl = fn' N cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xl:RHl:derivable_pt_abs (fn N) c lTemp:l = fn' N cfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsx:Rdelta:posrealh:Rxhinbxdelta:Boule x delta (x + h)c:RP:x + h < c < xBoule x delta cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xl:RHl:derivable_pt_abs (fn N) c lt:Boule x delta cBoule c' r cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xl:RHl:derivable_pt_abs (fn N) c lbc'rc:Boule c' r cl = fn' N cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xl:RHl:derivable_pt_abs (fn N) c lTemp:l = fn' N cfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsx:Rdelta:posrealh:RH1:x + h - x < deltaH2:- delta < x + h - xc:RH:x + h < cH0:c < xBoule x delta cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xl:RHl:derivable_pt_abs (fn N) c lt:Boule x delta cBoule c' r cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xl:RHl:derivable_pt_abs (fn N) c lbc'rc:Boule c' r cl = fn' N cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xl:RHl:derivable_pt_abs (fn N) c lTemp:l = fn' N cfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xl:RHl:derivable_pt_abs (fn N) c lt:Boule x delta cBoule c' r cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xl:RHl:derivable_pt_abs (fn N) c lbc'rc:Boule c' r cl = fn' N cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xl:RHl:derivable_pt_abs (fn N) c lTemp:l = fn' N cfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xl:RHl:derivable_pt_abs (fn N) c lbc'rc:Boule c' r cl = fn' N cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xl:RHl:derivable_pt_abs (fn N) c lTemp:l = fn' N cfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xl:RHl:derivable_pt_abs (fn N) c lbc'rc:Boule c' r cHl':derivable_pt_lim (fn N) c (fn' N c)l = fn' N cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xl:RHl:derivable_pt_abs (fn N) c lTemp:l = fn' N cfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> RN2, N1, N3:natN:=(N1 + N2 + N3)%nat:natc, l:RHl:derivable_pt_lim (fn N) c lHl':derivable_pt_lim (fn N) c (fn' N c)l = fn' N cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xl:RHl:derivable_pt_abs (fn N) c lTemp:l = fn' N cfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xl:RHl:derivable_pt_abs (fn N) c lTemp:l = fn' N cfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xl:RHl:derivable_pt_abs (fn N) c lTemp:l = fn' N cl = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xl:RHl:derivable_pt_abs (fn N) c lTemp:l = fn' N cderivable_pt (fn N) cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xl:RHl:derivable_pt_abs (fn N) c lTemp:l = fn' N cHl':derivable_pt (fn N) cl = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xl:RHl:derivable_pt_abs (fn N) c lTemp:l = fn' N cHl':derivable_pt (fn N) cl = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xl:RHl:derivable_pt_abs (fn N) c lTemp:l = fn' N cHl':derivable_pt (fn N) cl = derive_pt (fn N) c Hl'fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xl:RHl:derivable_pt_abs (fn N) c lTemp:l = fn' N cHl':derivable_pt (fn N) cfn N = fn Nfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xl:RHl:derivable_pt_abs (fn N) c lTemp:l = fn' N cl':RHl':derivable_pt_abs (fn N) c l'l = derive_pt (fn N) c (exist (fun l0 : R => derivable_pt_abs (fn N) c l0) l' Hl')fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xl:RHl:derivable_pt_abs (fn N) c lTemp:l = fn' N cHl':derivable_pt (fn N) cfn N = fn Nfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xl:RHl:derivable_pt_abs (fn N) c lTemp:l = fn' N cl':RHl':derivable_pt_abs (fn N) c l'l = l'fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xl:RHl:derivable_pt_abs (fn N) c lTemp:l = fn' N cl':RHl':derivable_pt_abs (fn N) c l'Main:l = l'l = derive_pt (fn N) c (exist (fun l0 : R => derivable_pt_abs (fn N) c l0) l' Hl')fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xl:RHl:derivable_pt_abs (fn N) c lTemp:l = fn' N cHl':derivable_pt (fn N) cfn N = fn Nfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xl:RHl:derivable_pt_abs (fn N) c lTemp:l = fn' N cl':RHl':derivable_pt_abs (fn N) c l'Main:l = l'l = derive_pt (fn N) c (exist (fun l0 : R => derivable_pt_abs (fn N) c l0) l' Hl')fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xl:RHl:derivable_pt_abs (fn N) c lTemp:l = fn' N cHl':derivable_pt (fn N) cfn N = fn Nfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:h < 0pr1:forall c0 : R, x + h < c0 < x -> derivable_pt (fn N) c0pr2:forall c0 : R, x + h < c0 < x -> derivable_pt id c0xh_x:x + h < xpr3:forall c0 : R, x + h <= c0 <= x -> continuity_pt (fn N) c0pr4:forall c0 : R, x + h <= c0 <= x -> continuity_pt id c0c:RP:x + h < c < xl:RHl:derivable_pt_abs (fn N) c lTemp:l = fn' N cHl':derivable_pt (fn N) cfn N = fn Nfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hh > 0fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hh_pos:h > 0Rabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hHyp:0 < hh > 0fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hHyp:0 = hh > 0fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hh_pos:h > 0Rabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hHyp:0 = hh > 0fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hh_pos:h > 0Rabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natsgn_h:0 <= hh_pos:h > 0Rabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0Rabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0forall c : R, x < c < x + h -> derivable_pt (fn N) cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c : R, x < c < x + h -> derivable_pt (fn N) cRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0c:Rc_encad:x < c < x + h{l : R | derivable_pt_abs (fn N) c l}fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c : R, x < c < x + h -> derivable_pt (fn N) cRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0c:Rc_encad:x < c < x + hBoule c' r cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c : R, x < c < x + h -> derivable_pt (fn N) cRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0c:Rc_encad:x < c < x + hBoule x delta cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0c:Rc_encad:x < c < x + ht:Boule x delta cBoule c' r cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c : R, x < c < x + h -> derivable_pt (fn N) cRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4H:x + h - x < deltaH0:- delta < x + h - xN1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0c:RH1:x < cH2:c < x + hBoule x delta cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0c:Rc_encad:x < c < x + ht:Boule x delta cBoule c' r cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c : R, x < c < x + h -> derivable_pt (fn N) cRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0c:Rc_encad:x < c < x + ht:Boule x delta cBoule c' r cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c : R, x < c < x + h -> derivable_pt (fn N) cRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c : R, x < c < x + h -> derivable_pt (fn N) cRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c : R, x < c < x + h -> derivable_pt (fn N) cforall c : R, x < c < x + h -> derivable_pt id cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c : R, x < c < x + h -> derivable_pt (fn N) cpr2:forall c : R, x < c < x + h -> derivable_pt id cRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c : R, x < c < x + h -> derivable_pt (fn N) cpr2:forall c : R, x < c < x + h -> derivable_pt id cRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c : R, x < c < x + h -> derivable_pt (fn N) cpr2:forall c : R, x < c < x + h -> derivable_pt id cxh_x:x < x + hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c : R, x < c < x + h -> derivable_pt (fn N) cpr2:forall c : R, x < c < x + h -> derivable_pt id cxh_x:x < x + hforall c : R, x <= c <= x + h -> continuity_pt (fn N) cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c : R, x < c < x + h -> derivable_pt (fn N) cpr2:forall c : R, x < c < x + h -> derivable_pt id cxh_x:x < x + hpr3:forall c : R, x <= c <= x + h -> continuity_pt (fn N) cRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hc:Rc_encad:x <= c <= x + hderivable_pt (fn N) cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c : R, x < c < x + h -> derivable_pt (fn N) cpr2:forall c : R, x < c < x + h -> derivable_pt id cxh_x:x < x + hpr3:forall c : R, x <= c <= x + h -> continuity_pt (fn N) cRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h = 0 -> Falsehinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hc:RH:x <= cH0:c <= x + hBoule c' r cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c : R, x < c < x + h -> derivable_pt (fn N) cpr2:forall c : R, x < c < x + h -> derivable_pt id cxh_x:x < x + hpr3:forall c : R, x <= c <= x + h -> continuity_pt (fn N) cRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h = 0 -> Falsehinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hc:RH:x <= cH0:c <= x + hBoule x delta cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h = 0 -> Falsehinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hc:RH:x <= cH0:c <= x + ht:Boule x delta cBoule c' r cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c : R, x < c < x + h -> derivable_pt (fn N) cpr2:forall c : R, x < c < x + h -> derivable_pt id cxh_x:x < x + hpr3:forall c : R, x <= c <= x + h -> continuity_pt (fn N) cRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h = 0 -> Falsehinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4H1:x + h - x < deltaH2:- delta < x + h - xN1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hc:RH:x <= cH0:c <= x + hBoule x delta cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h = 0 -> Falsehinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hc:RH:x <= cH0:c <= x + ht:Boule x delta cBoule c' r cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c : R, x < c < x + h -> derivable_pt (fn N) cpr2:forall c : R, x < c < x + h -> derivable_pt id cxh_x:x < x + hpr3:forall c : R, x <= c <= x + h -> continuity_pt (fn N) cRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h = 0 -> Falsehinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hc:RH:x <= cH0:c <= x + ht:Boule x delta cBoule c' r cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c : R, x < c < x + h -> derivable_pt (fn N) cpr2:forall c : R, x < c < x + h -> derivable_pt id cxh_x:x < x + hpr3:forall c : R, x <= c <= x + h -> continuity_pt (fn N) cRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c : R, x < c < x + h -> derivable_pt (fn N) cpr2:forall c : R, x < c < x + h -> derivable_pt id cxh_x:x < x + hpr3:forall c : R, x <= c <= x + h -> continuity_pt (fn N) cRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c : R, x < c < x + h -> derivable_pt (fn N) cpr2:forall c : R, x < c < x + h -> derivable_pt id cxh_x:x < x + hpr3:forall c : R, x <= c <= x + h -> continuity_pt (fn N) cforall c : R, x <= c <= x + h -> continuity_pt id cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c : R, x < c < x + h -> derivable_pt (fn N) cpr2:forall c : R, x < c < x + h -> derivable_pt id cxh_x:x < x + hpr3:forall c : R, x <= c <= x + h -> continuity_pt (fn N) cpr4:forall c : R, x <= c <= x + h -> continuity_pt id cRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c : R, x < c < x + h -> derivable_pt (fn N) cpr2:forall c : R, x < c < x + h -> derivable_pt id cxh_x:x < x + hpr3:forall c : R, x <= c <= x + h -> continuity_pt (fn N) cpr4:forall c : R, x <= c <= x + h -> continuity_pt id cRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hHc:(id (x + h) - id x) * derive_pt (fn N) c (pr1 c P) = (fn N (x + h) - fn N x) * derive_pt id c (pr2 c P)Rabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hHc:(id (x + h) - id x) * derive_pt (fn N) c (pr1 c P) = (fn N (x + h) - fn N x) * derive_pt id c (pr2 c P)h * derive_pt (fn N) c (pr1 c P) = fn N (x + h) - fn N xfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hHc:(id (x + h) - id x) * derive_pt (fn N) c (pr1 c P) = (fn N (x + h) - fn N x) * derive_pt id c (pr2 c P)Hc':h * derive_pt (fn N) c (pr1 c P) = fn N (x + h) - fn N xRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hHc:(id (x + h) - id x) * derive_pt (fn N) c (pr1 c P) = (fn N (x + h) - fn N x) * derive_pt id c (pr2 c P)(id (x + h) - id x) * derive_pt (fn N) c (pr1 c P) = fn N (x + h) - fn N xfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hHc:(id (x + h) - id x) * derive_pt (fn N) c (pr1 c P) = (fn N (x + h) - fn N x) * derive_pt id c (pr2 c P)Hc':h * derive_pt (fn N) c (pr1 c P) = fn N (x + h) - fn N xRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hHc:(id (x + h) - id x) * derive_pt (fn N) c (pr1 c P) = (fn N (x + h) - fn N x) * derive_pt id c (pr2 c P)(id (x + h) - id x) * derive_pt (fn N) c (pr1 c P) = (fn N (x + h) - fn N x) * derive_pt id c (pr2 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hHc:(id (x + h) - id x) * derive_pt (fn N) c (pr1 c P) = (fn N (x + h) - fn N x) * derive_pt id c (pr2 c P)Hc':h * derive_pt (fn N) c (pr1 c P) = fn N (x + h) - fn N xRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hHc:(id (x + h) - id x) * derive_pt (fn N) c (pr1 c P) = (fn N (x + h) - fn N x) * derive_pt id c (pr2 c P)Hc':h * derive_pt (fn N) c (pr1 c P) = fn N (x + h) - fn N xRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (fn N (x + h) - fn N x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (h * derive_pt (fn N) c (pr1 c P) - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (h * fn' N c - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs (h * (fn' N c - g x)) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs h * Rabs (fn' N c - g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hRabs (f (x + h) - fn N (x + h)) + Rabs (f x - fn N x) + Rabs h * Rabs (fn' N c - g x) < Rabs h * eps / 4 + Rabs (f x - fn N x) + Rabs h * Rabs (fn' N c - g x)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hRabs h * eps / 4 + Rabs (f x - fn N x) + Rabs h * Rabs (fn' N c - g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> Rabs (fn n (x + h) - f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + h(N >= N1)%natfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hRabs h * eps / 4 + Rabs (f x - fn N x) + Rabs h * Rabs (fn' N c - g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hRabs h * eps / 4 + Rabs (f x - fn N x) + Rabs h * Rabs (fn' N c - g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hRabs h * eps / 4 + Rabs (f x - fn N x) + Rabs h * Rabs (fn' N c - g x) < Rabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * Rabs (fn' N c - g x)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hRabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * Rabs (fn' N c - g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hRabs (f x - fn N x) < Rabs h * eps / 4fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hRabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * Rabs (fn' N c - g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> Rabs (fn n x - f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + h(N >= N2)%natfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hRabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * Rabs (fn' N c - g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hRabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * Rabs (fn' N c - g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hRabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * Rabs (fn' N c - g c + (g c - g x)) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hRabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * Rabs (fn' N c - g c + (g c - g x)) <= Rabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * Rabs (fn' N c - g c) + Rabs h * Rabs (g c - g x)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hRabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * Rabs (fn' N c - g c) + Rabs h * Rabs (g c - g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + h0 <= Rabs hfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hRabs (fn' N c - g c + (g c - g x)) <= Rabs (fn' N c - g c) + Rabs (g c - g x)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hRabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * Rabs (fn' N c - g c) + Rabs h * Rabs (g c - g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hRabs (fn' N c - g c + (g c - g x)) <= Rabs (fn' N c - g c) + Rabs (g c - g x)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hRabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * Rabs (fn' N c - g c) + Rabs h * Rabs (g c - g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hRabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * Rabs (fn' N c - g c) + Rabs h * Rabs (g c - g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hRabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * Rabs (fn' N c - g c) + Rabs h * Rabs (g c - g x) < Rabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * (eps / 8) + Rabs h * Rabs (g c - g x)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hRabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * (eps / 8) + Rabs h * Rabs (g c - g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + h0 < Rabs hfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hRabs (fn' N c - g c) < eps / 8fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hRabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * (eps / 8) + Rabs h * Rabs (g c - g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hRabs (fn' N c - g c) < eps / 8fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hRabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * (eps / 8) + Rabs h * Rabs (g c - g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + h(N3 <= N)%natfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hBoule c' r cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hRabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * (eps / 8) + Rabs h * Rabs (g c - g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hBoule c' r cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hRabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * (eps / 8) + Rabs h * Rabs (g c - g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hBoule x delta cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + ht:Boule x delta cBoule c' r cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hRabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * (eps / 8) + Rabs h * Rabs (g c - g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RH:x < cH0:c < x + hBoule x delta cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + ht:Boule x delta cBoule c' r cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hRabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * (eps / 8) + Rabs h * Rabs (g c - g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4H1:x + h - x < deltaH2:- delta < x + h - xN1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RH:x < cH0:c < x + hBoule x delta cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + ht:Boule x delta cBoule c' r cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hRabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * (eps / 8) + Rabs h * Rabs (g c - g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + ht:Boule x delta cBoule c' r cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hRabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * (eps / 8) + Rabs h * Rabs (g c - g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hRabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * (eps / 8) + Rabs h * Rabs (g c - g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hRabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * (eps / 8) + Rabs h * Rabs (g c - g x) < Rabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * (eps / 8) + Rabs h * (eps / 8)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hRabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * (eps / 8) + Rabs h * (eps / 8) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + h0 < Rabs hfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + heps / 8 + Rabs (g c - g x) < eps / 8 + eps / 8fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hRabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * (eps / 8) + Rabs h * (eps / 8) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + heps / 8 + Rabs (g c - g x) < eps / 8 + eps / 8fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hRabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * (eps / 8) + Rabs h * (eps / 8) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : R, D_x no_cond x x0 /\ R_dist x0 x < delta1 -> R_dist (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hno_cond cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : R, D_x no_cond x x0 /\ R_dist x0 x < delta1 -> R_dist (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hx <> cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : R, D_x no_cond x x0 /\ R_dist x0 x < delta1 -> R_dist (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hR_dist c x < delta1fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hRabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * (eps / 8) + Rabs h * (eps / 8) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : R, D_x no_cond x x0 /\ R_dist x0 x < delta1 -> R_dist (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hx <> cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : R, D_x no_cond x x0 /\ R_dist x0 x < delta1 -> R_dist (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hR_dist c x < delta1fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hRabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * (eps / 8) + Rabs h * (eps / 8) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : R, D_x no_cond x x0 /\ R_dist x0 x < delta1 -> R_dist (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hR_dist c x < delta1fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hRabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * (eps / 8) + Rabs h * (eps / 8) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : R, D_x no_cond x x0 /\ R_dist x0 x < delta1 -> R_dist (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hR_dist c x < Rabs hfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : R, D_x no_cond x x0 /\ R_dist x0 x < delta1 -> R_dist (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hRabs h < delta1fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hRabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * (eps / 8) + Rabs h * (eps / 8) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : R, D_x no_cond x x0 /\ R_dist x0 x < delta1 -> R_dist (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hc - x < Rabs hfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : R, D_x no_cond x x0 /\ R_dist x0 x < delta1 -> R_dist (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + h- Rabs h < c - xfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : R, D_x no_cond x x0 /\ R_dist x0 x < delta1 -> R_dist (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hRabs h < delta1fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hRabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * (eps / 8) + Rabs h * (eps / 8) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : R, D_x no_cond x x0 /\ R_dist x0 x < delta1 -> R_dist (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + h- Rabs h < c - xfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : R, D_x no_cond x x0 /\ R_dist x0 x < delta1 -> R_dist (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hRabs h < delta1fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hRabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * (eps / 8) + Rabs h * (eps / 8) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : R, D_x no_cond x x0 /\ R_dist x0 x < delta1 -> R_dist (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + h- Rabs h <= 0fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : R, D_x no_cond x x0 /\ R_dist x0 x < delta1 -> R_dist (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + h0 < c - xfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : R, D_x no_cond x x0 /\ R_dist x0 x < delta1 -> R_dist (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hRabs h < delta1fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hRabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * (eps / 8) + Rabs h * (eps / 8) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : R, D_x no_cond x x0 /\ R_dist x0 x < delta1 -> R_dist (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + h0 < c - xfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : R, D_x no_cond x x0 /\ R_dist x0 x < delta1 -> R_dist (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hRabs h < delta1fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hRabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * (eps / 8) + Rabs h * (eps / 8) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : R, D_x no_cond x x0 /\ R_dist x0 x < delta1 -> R_dist (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hRabs h < delta1fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hRabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * (eps / 8) + Rabs h * (eps / 8) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsx, c':Rr:posrealdelta1:Rdelta1_pos:delta1 > 0delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rxhinbxdelta:Boule x delta (x + h)Rabs h < delta1fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hRabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * (eps / 8) + Rabs h * (eps / 8) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsx, c':Rr:posrealdelta1:Rdelta1_pos:delta1 > 0delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:RP':Boule x {| pos := delta1; cond_pos := delta1_pos |} (x + h)Rabs h < delta1fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hRabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * (eps / 8) + Rabs h * (eps / 8) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hRabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * (eps / 8) + Rabs h * (eps / 8) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hRabs h * eps / 4 + (Rabs h * eps / 4 + Rabs h * (eps / 8 + eps / 8)) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hRabs h * (eps / 4 + eps / 4 + eps / 8 + eps / 8) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + h0 < Rabs hfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + heps / 4 + eps / 4 + eps / 8 + eps / 8 < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + heps / 4 + eps / 4 + eps / 8 + eps / 8 < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hl:RHl:derivable_pt_abs (fn N) c lfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hl:RHl:derivable_pt_abs (fn N) c ll = fn' N cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hl:RHl:derivable_pt_abs (fn N) c lTemp:l = fn' N cfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hl:RHl:derivable_pt_abs (fn N) c lBoule c' r cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hl:RHl:derivable_pt_abs (fn N) c lbc'rc:Boule c' r cl = fn' N cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hl:RHl:derivable_pt_abs (fn N) c lTemp:l = fn' N cfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hl:RHl:derivable_pt_abs (fn N) c lBoule x delta cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hl:RHl:derivable_pt_abs (fn N) c lt:Boule x delta cBoule c' r cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hl:RHl:derivable_pt_abs (fn N) c lbc'rc:Boule c' r cl = fn' N cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hl:RHl:derivable_pt_abs (fn N) c lTemp:l = fn' N cfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsx:Rdelta:posrealh:Rxhinbxdelta:Boule x delta (x + h)c:RP:x < c < x + hBoule x delta cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hl:RHl:derivable_pt_abs (fn N) c lt:Boule x delta cBoule c' r cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hl:RHl:derivable_pt_abs (fn N) c lbc'rc:Boule c' r cl = fn' N cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hl:RHl:derivable_pt_abs (fn N) c lTemp:l = fn' N cfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsx:Rdelta:posrealh:RH1:x + h - x < deltaH2:- delta < x + h - xc:RH:x < cH0:c < x + hBoule x delta cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hl:RHl:derivable_pt_abs (fn N) c lt:Boule x delta cBoule c' r cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hl:RHl:derivable_pt_abs (fn N) c lbc'rc:Boule c' r cl = fn' N cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hl:RHl:derivable_pt_abs (fn N) c lTemp:l = fn' N cfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hl:RHl:derivable_pt_abs (fn N) c lt:Boule x delta cBoule c' r cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hl:RHl:derivable_pt_abs (fn N) c lbc'rc:Boule c' r cl = fn' N cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hl:RHl:derivable_pt_abs (fn N) c lTemp:l = fn' N cfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hl:RHl:derivable_pt_abs (fn N) c lbc'rc:Boule c' r cl = fn' N cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hl:RHl:derivable_pt_abs (fn N) c lTemp:l = fn' N cfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hl:RHl:derivable_pt_abs (fn N) c lbc'rc:Boule c' r cHl':derivable_pt_lim (fn N) c (fn' N c)l = fn' N cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hl:RHl:derivable_pt_abs (fn N) c lTemp:l = fn' N cfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> RN2, N1, N3:natN:=(N1 + N2 + N3)%nat:natc, l:RHl:derivable_pt_lim (fn N) c lHl':derivable_pt_lim (fn N) c (fn' N c)l = fn' N cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hl:RHl:derivable_pt_abs (fn N) c lTemp:l = fn' N cfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hl:RHl:derivable_pt_abs (fn N) c lTemp:l = fn' N cfn' N c = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hl:RHl:derivable_pt_abs (fn N) c lTemp:l = fn' N cl = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hl:RHl:derivable_pt_abs (fn N) c lTemp:l = fn' N cderivable_pt (fn N) cfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hl:RHl:derivable_pt_abs (fn N) c lTemp:l = fn' N cHl':derivable_pt (fn N) cl = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hl:RHl:derivable_pt_abs (fn N) c lTemp:l = fn' N cHl':derivable_pt (fn N) cl = derive_pt (fn N) c (pr1 c P)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hl:RHl:derivable_pt_abs (fn N) c lTemp:l = fn' N cHl':derivable_pt (fn N) cl = derive_pt (fn N) c Hl'fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hl:RHl:derivable_pt_abs (fn N) c lTemp:l = fn' N cHl':derivable_pt (fn N) cfn N = fn Nfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hl:RHl:derivable_pt_abs (fn N) c lTemp:l = fn' N cl':RHl':derivable_pt_abs (fn N) c l'l = derive_pt (fn N) c (exist (fun l0 : R => derivable_pt_abs (fn N) c l0) l' Hl')fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hl:RHl:derivable_pt_abs (fn N) c lTemp:l = fn' N cHl':derivable_pt (fn N) cfn N = fn Nfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hl:RHl:derivable_pt_abs (fn N) c lTemp:l = fn' N cl':RHl':derivable_pt_abs (fn N) c l'l = l'fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hl:RHl:derivable_pt_abs (fn N) c lTemp:l = fn' N cl':RHl':derivable_pt_abs (fn N) c l'Main:l = l'l = derive_pt (fn N) c (exist (fun l0 : R => derivable_pt_abs (fn N) c l0) l' Hl')fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hl:RHl:derivable_pt_abs (fn N) c lTemp:l = fn' N cHl':derivable_pt (fn N) cfn N = fn Nfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hl:RHl:derivable_pt_abs (fn N) c lTemp:l = fn' N cl':RHl':derivable_pt_abs (fn N) c l'Main:l = l'l = derive_pt (fn N) c (exist (fun l0 : R => derivable_pt_abs (fn N) c l0) l' Hl')fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hl:RHl:derivable_pt_abs (fn N) c lTemp:l = fn' N cHl':derivable_pt (fn N) cfn N = fn Nfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:nath_pos:h > 0pr1:forall c0 : R, x < c0 < x + h -> derivable_pt (fn N) c0pr2:forall c0 : R, x < c0 < x + h -> derivable_pt id c0xh_x:x < x + hpr3:forall c0 : R, x <= c0 <= x + h -> continuity_pt (fn N) c0pr4:forall c0 : R, x <= c0 <= x + h -> continuity_pt id c0c:RP:x < c < x + hl:RHl:derivable_pt_abs (fn N) c lTemp:l = fn' N cHl':derivable_pt (fn N) cfn N = fn Nfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs ((f (x + h) - f x) / h - g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs (/ h * (f (x + h) - f x - h * g x)) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * eps/ h * (f (x + h) - f x - h * g x) = (f (x + h) - f x) / h - g xfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * eps/ Rabs h * Rabs (f (x + h) - f x - h * g x) < epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsh <> 0fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * eps/ h * (f (x + h) - f x - h * g x) = (f (x + h) - f x) / h - g xfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * eps/ Rabs h * Rabs (f (x + h) - f x - h * g x) < / Rabs h * (Rabs h * eps)fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * eps/ Rabs h * (Rabs h * eps) = epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsh <> 0fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * eps/ h * (f (x + h) - f x - h * g x) = (f (x + h) - f x) / h - g xfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * eps0 < / Rabs hfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs (f (x + h) - f x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * eps/ Rabs h * (Rabs h * eps) = epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsh <> 0fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * eps/ h * (f (x + h) - f x - h * g x) = (f (x + h) - f x) / h - g xfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs (f (x + h) - f x - h * g x) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * eps/ Rabs h * (Rabs h * eps) = epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsh <> 0fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * eps/ h * (f (x + h) - f x - h * g x) = (f (x + h) - f x) / h - g xfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsRabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * eps/ Rabs h * (Rabs h * eps) = epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsh <> 0fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * eps/ h * (f (x + h) - f x - h * g x) = (f (x + h) - f x) / h - g xfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * eps/ Rabs h * (Rabs h * eps) = epsfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsh <> 0fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * eps/ h * (f (x + h) - f x - h * g x) = (f (x + h) - f x) / h - g xfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsh <> 0fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * eps/ h * (f (x + h) - f x - h * g x) = (f (x + h) - f x) / h - g xfn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * eps/ h * (f (x + h) - f x - h * g x) = (f (x + h) - f x) / h - g xassumption. Qed.fn, fn':nat -> R -> Rf, g:R -> Rx, c':Rr:posrealxinb:Boule c' r xDfn_eq_fn':forall (y : R) (n : nat), Boule c' r y -> derivable_pt_lim (fn n) y (fn' n y)fn'_CVU_g:CVU fn' g c' reps:Reps_pos:0 < epseps_8_pos:0 < eps / 8delta1:Rdelta1_pos:delta1 > 0g_cont:forall x0 : Base R_met, D_x no_cond x x0 /\ dist R_met x0 x < delta1 -> dist R_met (g x0) (g x) < eps / 8delta:posrealPdelta:forall y : R, Boule x delta y -> Boule c' r y /\ Boule x {| pos := delta1; cond_pos := delta1_pos |} yh:Rhpos:h <> 0hinbdelta:Rabs h < deltaeps'_pos:0 < Rabs h * eps / 4N2:natfnx_CV_fx:forall n : nat, (n >= N2)%nat -> R_dist (fn n x) (f x) < Rabs h * eps / 4xhinbxdelta:Boule x delta (x + h)N1:natfnxh_CV_fxh:forall n : nat, (n >= N1)%nat -> R_dist (fn n (x + h)) (f (x + h)) < Rabs h * eps / 4N3:natfn'c_CVU_gc:forall (n : nat) (y : R), (N3 <= n)%nat -> Boule c' r y -> Rabs (g y - fn' n y) < eps / 8N:=(N1 + N2 + N3)%nat:natMain:Rabs (f (x + h) - fn N (x + h) - (f x - fn N x) + (fn N (x + h) - fn N x - h * g x)) < Rabs h * epsh <> 0